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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">d</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56"/><span class="lab"><font color="Green" title="E39">A1</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">and </span><br/><a NAME="E2:56"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c2">b</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">c</font>, <font color="Olive" title="b2">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">d</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:56"/><span class="lab"><font color="Green" title="E41">A3</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T27" target="_self" title="WSIERP_1:th.27">Th34</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">k</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1;<br/>
<a NAME="E5:56"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T32" title="INT_1:th.32">INT_1:32</a>, <a class="ref" href="xxreal_0.html#T31" title="XXREAL_0:th.31">XXREAL_0:31</a></span>;<br/>
<a NAME="E6:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xreal_1.html#T77" title="XREAL_1:th.77">XREAL_1:77</a></span>;<br/>
<a NAME="E7:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E8:56"/>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T32" title="INT_1:th.32">INT_1:32</a>, <a class="ref" href="xxreal_0.html#T31" title="XXREAL_0:th.31">XXREAL_0:31</a></span>;<br/>
<a NAME="E9:56"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T187" title="XCMPLX_1:th.187">XCMPLX_1:187</a></span>;<br/>
<a NAME="E10:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="xreal_1.html#T77" title="XREAL_1:th.77">XREAL_1:77</a></span>;<br/>
<a NAME="E11:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">e</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c7">d</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">a</font></span>)</span>)</span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">n</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="#E7:56"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E13:56"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c6">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E14:56"/>
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E41">A3</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/>


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