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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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"> ( <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  implies  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">a9</font> = 1, <font color="Maroon" title="c4">09</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">assume </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</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">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:10"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 ex <font color="Olive" title="b1">i0</font>, <font color="Olive" title="b2">j0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">and </span><br/><a NAME="E5:10"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 for <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/><font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T9" target="_self" onmouseover="rs('euler_1/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">i0</font>, <font color="Maroon" title="c8">j0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E7:10"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c8">j0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:10"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<a NAME="E9:10"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 for <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/><font color="Olive" title="b1">c</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">b</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"> (  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies <font color="Maroon" title="c9">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c9">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c11">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:10_1"/>
<font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">t2</font> = <font color="Maroon" title="c11">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">j0</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">t1</font> = <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i0</font></span>)</span>;<br/>
<a NAME="E5:10_1"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="nat_d.html#T2" onmouseover="rs('nat_d/T2')" onmouseout="rh()">NAT_D:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">c</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">j0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:10_1"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</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"> contradiction</span><br/>

<a NAME="E8:10_1"/>
<font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="nat_d.html#T1" onmouseover="rs('nat_d/T1')" onmouseout="rh()">NAT_D:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E7:10_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E20">A8</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>
<a NAME="E10:10"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_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="E1:10_2_1"/>
( <font color="Maroon" title="c2">n</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="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="10_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_2_1_1"/>
<font color="Maroon" title="c2">n</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="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:10_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" onmouseover="rs('nat_d/T6')" onmouseout="rh()">NAT_D:6</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="10_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_2_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font></span><br/><div class="add"><a NAME="E2:10_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">09</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c3">a9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E3:10_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font> <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="txt" href="#E9:10"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/><a NAME="E4:10_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="nat_d.html#T2" onmouseover="rs('nat_d/T2')" onmouseout="rh()">NAT_D:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_2_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T3" onmouseover="rs('nat_d/T3')" onmouseout="rh()">NAT_D:3</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>
<a NAME="E11:10"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_3_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="E1:10_3_1"/>
( <font color="Maroon" title="c1">m</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">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="10_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_3_1_1"/>
<font color="Maroon" title="c1">m</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="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:10_3_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" onmouseover="rs('nat_d/T6')" onmouseout="rh()">NAT_D:6</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="10_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_3_1_2"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font></span><br/><div class="add"><a NAME="E2:10_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c4">09</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E3:10_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">a</font> <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="txt" href="#E9:10"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>;<br/><a NAME="E4:10_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="nat_d.html#T2" onmouseover="rs('nat_d/T2')" onmouseout="rh()">NAT_D:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_3_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T3" onmouseover="rs('nat_d/T3')" onmouseout="rh()">NAT_D:3</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>
<a NAME="E12:10"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<a NAME="E13:10"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="#E2:10"><span class="lab"><font color="Green" title="E13">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="E14:10"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">b</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E15:10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c8">j0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="nat_1.html#T15" onmouseover="rs('nat_1/T15')" onmouseout="rh()">NAT_1:15</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">j0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">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="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">j0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>

;<br/>
<a NAME="E17:10"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <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="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">j0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E18:10"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
