<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</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">n0</font>, <font color="Olive" title="b2">m0</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>  st <font color="Olive" title="b1">n0</font>,<font color="Olive" title="b2">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  holds <br/><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Olive" title="b1">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Olive" title="b2">m0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n0</font>, <font color="Maroon" title="c3">m0</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"> ( <font color="Maroon" title="c2">n0</font>,<font color="Maroon" title="c3">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  implies <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">n0</font>,<font color="Maroon" title="c3">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</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="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T8" onmouseover="rs('wsierp_1/T8')" onmouseout="rh()">WSIERP_1:8</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">n0</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">m0</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c3">m0</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="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="wsierp_1.html#T8" onmouseover="rs('wsierp_1/T8')" onmouseout="rh()">WSIERP_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="wsierp_1.html#T8" onmouseover="rs('wsierp_1/T8')" onmouseout="rh()">WSIERP_1:8</a></span>
;<br/> <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="19_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2"/>
<font color="Maroon" title="c1">k</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"> <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">k9</font> = <font color="Maroon" title="c1">k</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" onmouseover="rs('int_2/T5')" onmouseout="rh()">INT_2:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">b1</font> = <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font>, <font color="Maroon" title="c7">b2</font> = <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" onmouseover="rs('int_2/T5')" onmouseout="rh()">INT_2:5</a></span>;<br/><a NAME="E5:19_1_2"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</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; <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" onmouseover="rs('int_2/T5')" onmouseout="rh()">INT_2:5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E7:19_1_2"/>
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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="c9">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_2_1"/>
<font color="Maroon" title="c9">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</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="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p9</font> = <font color="Maroon" title="c9">p</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> ;<br/>
<span class="lab"><font color="Green" title="E22">A3</font></span>: <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> = 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">k9</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T28" onmouseover="rs('nat_3/T28')" onmouseout="rh()">NAT_3:28</a></span>
;<br/>
<a NAME="E4:19_1_2_1"/>
<font color="Maroon" title="c2">n0</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="int_2.html#D3" onmouseover="rs('int_2/D3')" onmouseout="rh()">INT_2:def 3</a></span>;<br/>
<a NAME="E5:19_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">p9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> 1 <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="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<a NAME="E6:19_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A4</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="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font></span>)</span>) <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_3.html#T21" onmouseover="rs('nat_3/T21')" onmouseout="rh()">NAT_3:21</a></span>;<br/>
<span class="lab"><font color="Green" title="E24">A5</font></span>: <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T28" onmouseover="rs('nat_3/T28')" onmouseout="rh()">NAT_3:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">k9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">k9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c4">k9</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_3"><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="E1:19_1_2_1_3"/>
( <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_1_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T15" onmouseover="rs('xxreal_0/T15')" onmouseout="rh()">XXREAL_0:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_2_1_3_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><a NAME="E2:19_1_2_1_3_1"/><span class="kw">then </span>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n0</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/><span class="kw">hence </span><a NAME="E3:19_1_2_1_3_1"/>
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_2_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:19_1_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E1:19_1_2_1_3_1"><span class="lab"><font color="Green" title="E25">A6</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2_1_3_2"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><a NAME="E2:19_1_2_1_3_2"/><span class="kw">then </span>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m0</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/><span class="kw">hence </span><a NAME="E3:19_1_2_1_3_2"/>
<font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_2_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E7:19_1_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E1:19_1_2_1_3_2"><span class="lab"><font color="Green" title="E25">A7</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:19_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T21" onmouseover="rs('nat_4/T21')" onmouseout="rh()">NAT_4:21</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>

</div>
