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<span class="kw">let </span><font color="Maroon" title="c1">a</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">z</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">z</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">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</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"> 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">z</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">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/>



<a NAME="E1:60"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <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="nat_d.html#T13" onmouseover="rs('nat_d/T13')" onmouseout="rh()">NAT_D:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A2</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_6.html#T7" onmouseover="rs('int_6/T7')" onmouseout="rh()">INT_6:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3

<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T9" target="_self" onmouseover="rs('number02/T9')" onmouseout="rh()">Th9</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="number02.html#T39" target="_self" onmouseover="rs('number02/T39')" onmouseout="rh()">Th39</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" onmouseover="rs('nat_d/T25')" onmouseout="rh()">NAT_D:25</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:60"/>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">z</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">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <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="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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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