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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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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> <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="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p2">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span><br/>

<font color="Maroon" title="c2">a</font> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font>

<br/>.= 
<span class="p1">(<span class="default"><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> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p2">(<span class="default"><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></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="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T35" onmouseover="rs('newton03/T35')" onmouseout="rh()">NEWTON03:35</a>, <a class="ref" href="newton05.html#T87" target="_self" onmouseover="rs('newton05/T87')" onmouseout="rh()">OPC</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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"><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> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></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="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T35" onmouseover="rs('newton03/T35')" onmouseout="rh()">NEWTON03:35</a>, <a class="ref" href="newton05.html#T87" target="_self" onmouseover="rs('newton05/T87')" onmouseout="rh()">OPC</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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="nat_1.html#K2" title="NAT_1:func.2">*</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"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T87" target="_self" onmouseover="rs('newton05/T87')" onmouseout="rh()">OPC</a></span>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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> <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="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p2">(<span class="default"><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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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> <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"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</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"><span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#T35" onmouseover="rs('newton03/T35')" onmouseout="rh()">NEWTON03:35</a>, <a class="ref" href="newton05.html#T87" target="_self" onmouseover="rs('newton05/T87')" onmouseout="rh()">OPC</a></span>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p3">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton05.html#T87" target="_self" onmouseover="rs('newton05/T87')" onmouseout="rh()">OPC</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:127"/>
<font color="Maroon" title="c2">a</font> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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> <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="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <span class="p2">(<span class="default"><a href="newton05.html#K4" title="NEWTON05:func.4">Oddity</a> <font color="Maroon" title="c3">b</font></span>)</span></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"> verum</span><br/>


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