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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">  for <font color="Olive" title="b1">c</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">c</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</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"> <font color="Maroon" title="c4">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)</span><br/>



<a NAME="E1:66"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#D1" onmouseover="rs('newton03/D1')" onmouseout="rh()">NEWTON03:def 1</a></span>;<br/>
<a NAME="E2:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
( <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">a</font> &amp;  not <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton03.html#D7" onmouseover="rs('newton03/D7')" onmouseout="rh()">NEWTON03:def 7</a></span>;<br/>
<a NAME="E3:66"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
( <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">b</font> &amp;  not <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <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="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="newton03.html#D7" onmouseover="rs('newton03/D7')" onmouseout="rh()">NEWTON03:def 7</a></span>;<br/>
<a NAME="E4:66"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:66_1"/>
( <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_1"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</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"> <font color="Maroon" title="c4">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)</span><br/><div class="add"><a NAME="E2:66_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B2</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:66_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>;<br/><a NAME="E4:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>;<br/><a NAME="E5:66_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B3</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <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="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/><a NAME="E6:66_1_1"/>
 not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="int_2.html#T16" onmouseover="rs('int_2/T16')" onmouseout="rh()">INT_2:16</a></span>;<br/><a NAME="E7:66_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="newton.html#T50" onmouseover="rs('newton/T50')" onmouseout="rh()">NEWTON:50</a></span>;<br/><a NAME="E8:66_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span> <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> <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="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T34" onmouseover="rs('int_2/T34')" onmouseout="rh()">INT_2:34</a></span>;<br/><span class="kw">hence </span><a NAME="E9:66_1_1"/>
<font color="Maroon" title="c4">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:66_1_1"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="txt" href="#E5:66_1_1"><span class="lab"><font color="Green" title="E31">B3</font></span></a>, <a class="ref" href="newton03.html#D7" onmouseover="rs('newton03/D7')" onmouseout="rh()">NEWTON03:def 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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_2"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</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"> <font color="Maroon" title="c4">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)</span><br/><div class="add"><a NAME="E2:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B2</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:66_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>;<br/><a NAME="E4:66_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>;<br/><a NAME="E5:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B3</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <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="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="int_2.html#D2" onmouseover="rs('int_2/D2')" onmouseout="rh()">INT_2:def 2</a></span>;<br/><a NAME="E6:66_1_2"/>
 not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="int_2.html#T16" onmouseover="rs('int_2/T16')" onmouseout="rh()">INT_2:16</a></span>;<br/><a NAME="E7:66_1_2"/><span class="kw">then </span>
 not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="newton.html#T50" onmouseover="rs('newton/T50')" onmouseout="rh()">NEWTON:50</a></span>;<br/><a NAME="E8:66_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">c</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span> <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> <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="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T34" onmouseover="rs('int_2/T34')" onmouseout="rh()">INT_2:34</a></span>;<br/><span class="kw">hence </span><a NAME="E9:66_1_2"/>
<font color="Maroon" title="c4">c</font> <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="newton03.html#K2" title="NEWTON03:func.2">gcd</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="newton03.html#K4" title="NEWTON03:func.4">|-count</a> <font color="Maroon" title="c3">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:66_1_2"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="txt" href="#E5:66_1_2"><span class="lab"><font color="Green" title="E31">B3</font></span></a>, <a class="ref" href="newton03.html#D7" onmouseover="rs('newton03/D7')" onmouseout="rh()">NEWTON03:def 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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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