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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> implies  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:60"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E3:60"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E4:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T14" target="_self" title="WSIERP_1:th.14">Th19</a></span>;<br/>
<a NAME="E5:60"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E6:60"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="wsierp_1.html#T15" target="_self" title="WSIERP_1:th.15">Th20</a>, <a class="ref" href="newton.html#T57" title="NEWTON:th.57">NEWTON:57</a></span>;<br/>
<a NAME="E7:60"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T12" target="_self" title="WSIERP_1:th.12">Th17</a></span>;<br/>
<a NAME="E8:60"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="wsierp_1.html#T29" target="_self" title="WSIERP_1:th.29">Th36</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:60"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:60_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:60_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:60_1"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">d1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E6:60_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><div><span class="lab"><font color="Green" title="E49">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E2:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A12</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/><a NAME="E3:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:60_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E53">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c10">e1</font> = <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:60_1_1_1"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a1</font>, <font color="Maroon" title="c12">c1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:60_1_1_1"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">a1</font>
 <span class="kw">and </span><br/><a NAME="E5:60_1_1_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E6:60_1_1_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">c1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T23" title="INT_2:th.23">INT_2:23</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c11">a1</font>, <font color="Maroon" title="c14">c1</font> = <font color="Maroon" title="c12">c1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E2:60_1_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E4:60_1_1_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="wsierp_1.html#E23"><span class="lab"><font color="Green" title="E20">Lm6</font></span></a></span>;<br/><font color="Maroon" title="c13">a1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">d1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E4:60_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E4:60_1_1_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
;<br/><a NAME="E9:60_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A19</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c13">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E10:60_1_1_1"/>
<font color="Maroon" title="c13">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/><a NAME="E11:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_1_1"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="wsierp_1.html#T15" target="_self" title="WSIERP_1:th.15">Th20</a>, <a class="ref" href="newton.html#T57" title="NEWTON:th.57">NEWTON:57</a></span>;<br/><a NAME="E12:60_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c1</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T12" target="_self" title="WSIERP_1:th.12">Th17</a></span>;<br/><a NAME="E13:60_1_1_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><a NAME="E14:60_1_1_1"/>
<font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A22</font></span>: <font color="Maroon" title="c14">c1</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E10:60"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="prepower.html#T8" title="PREPOWER:th.8">PREPOWER:8</a></span>
<br/>.= 
<font color="Maroon" title="c7">g</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1_1_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
;<br/><a NAME="E16:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c14">c1</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E17:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60_1_1_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="newton.html#T49" title="NEWTON:th.49">NEWTON:49</a></span>;<br/><span class="kw">hence </span><a NAME="E18:60_1_1_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E15:60_1_1_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a></span>; <a class="txt" 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="hs2(this)" href="javascript:()" title="60_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E2:60_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E6:60_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E2:60_1_1"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/><a NAME="E3:60_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="newton.html#T52" title="NEWTON:th.52">NEWTON:52</a></span>;<br/><a NAME="E4:60_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A24</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E5:60_1_1_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E2:60_1_1_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:60_1_1_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a></span>; <a class="txt" 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><span class="kw">hence </span><a NAME="E6:60_1_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1"><span class="lab"><font color="Green" title="E53">A14</font></span></a></span>; <a class="txt" 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="hs2(this)" href="javascript:()" title="60_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E2:60_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/><a NAME="E3:60_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="newton.html#T52" title="NEWTON:th.52">NEWTON:52</a></span>;<br/><a NAME="E4:60_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E5:60_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E2:60_1_2"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:60_1_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_2"><span class="lab"><font color="Green" title="E51">A26</font></span></a></span>; <a class="txt" 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><span class="kw">hence </span><a NAME="E9:60_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:60_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a></span>; <a class="txt" 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="hs2(this)" href="javascript:()" title="60_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E2:60_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><a NAME="E3:60_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E4:60_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T15" target="_self" title="WSIERP_1:th.15">Th20</a></span>;<br/><a NAME="E5:60_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A29</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><a NAME="E6:60_2"/>
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_2"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E7:60_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T15" target="_self" title="WSIERP_1:th.15">Th20</a></span>;<br/><a NAME="E8:60_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:60_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E5:60_2"><span class="lab"><font color="Green" title="E47">A29</font></span></a></span>; <a class="txt" 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>
<span class="kw">hence </span><a NAME="E14:60"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
