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

<span class="kw">let </span><font color="Maroon" title="c1">i</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">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">n</font> or <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> or <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">n</font> &amp;  not $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> &amp;  not $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> ) );<br/>
<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c1">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">n</font> or <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> or <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E3:98"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E4:98"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
(  not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font> &amp;  not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:98"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E55">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">p9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:98"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p9</font>]
 <span class="kw">and </span><br/><a NAME="E8:98"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
 for <font color="Olive" title="b1">p1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] holds <br/><font color="Maroon" title="c4">p9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E56">A4</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">n</font> &amp;  not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 &amp;  not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font> );<br/>
<a NAME="E9:98"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E57">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:98"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">m</font>]
 <span class="kw">and </span><br/><a NAME="E12:98"/><span class="lab"><font color="Green" title="E61">A9</font></span>: 
 for <font color="Olive" title="b1">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m1</font>] holds <br/><font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m1</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E59">A7</font></span></a>);<br/></span>
<a NAME="E13:98"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
<font color="Maroon" title="c5">m</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="txt" href="#E11:98"><span class="lab"><font color="Green" title="E60">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:98"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">and </span><br/><a NAME="E16:98"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
 not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">and </span><br/><a NAME="E17:98"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E60">A8</font></span></a></span>;<br/>
<a NAME="E18:98"/><span class="lab"><font color="Green" title="E66">A14</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font>;<br/>
<span class="kw">assume </span><a NAME="E1:98_1"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">p9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:98_1"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
<font color="Maroon" title="c4">p9</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="txt" href="#E7:98"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E3:98_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p9</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p9</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/>
<a NAME="E4:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A17</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</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="c4">p9</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p9</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
;<br/>
<a NAME="E5:98_1"/>
<font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/>
<a NAME="E6:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A18</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:98_1"/><span class="lab"><font color="Green" title="E71">A19</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p9</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p9</font></span>)</span>
 
<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="E8:98_1"/>
<font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p9</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p9</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<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="E9:98_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A20</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:98"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="#E4:98_1"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="nat_d.html#T10" title="NAT_D:th.10">NAT_D:10</a></span>;<br/>
<a NAME="E10:98_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p9</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p9</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_1"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/>
<a NAME="E11:98_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E7:98_1"><span class="lab"><font color="Green" title="E71">A19</font></span></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:98_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="txt" href="#E6:98_1"><span class="lab"><font color="Green" title="E70">A18</font></span></a>, <a class="txt" href="#E9:98_1"><span class="lab"><font color="Green" title="E72">A20</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>
<a NAME="E19:98"/><span class="lab"><font color="Green" title="E67">A21</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:98_2"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p1</font> being    <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"> such that </span><br/><a NAME="E3:98_2"/><span class="lab"><font color="Green" title="E68">A22</font></span>: 
<font color="Maroon" title="c7">p1</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E4:98_2"/><span class="lab"><font color="Green" title="E69">A23</font></span>: 
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></span>;<br/>
<a NAME="E5:98_2"/><span class="lab"><font color="Green" title="E70">A24</font></span>: 
<font color="Maroon" title="c7">p1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E6:98_2"/><span class="lab"><font color="Green" title="E71">A25</font></span>: 
 not <font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:98_2_1"/>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:98_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:98_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E3:98_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</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>
<a NAME="E7:98_2"/>
<font color="Maroon" title="c7">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/>
<a NAME="E8:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A26</font></span>: 
 not <font color="Maroon" title="c4">p9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:98"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:98_2"/><span class="lab"><font color="Green" title="E73">A27</font></span>: 
<font color="Maroon" title="c7">p1</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="#E3:98_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:98_2"/><span class="lab"><font color="Green" title="E74">A28</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:98"><span class="lab"><font color="Green" title="E63">A11</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="E12:98_2"/>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="txt" href="#E11:98_2"><span class="lab"><font color="Green" title="E74">A28</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E13:98_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="txt" href="#E3:98_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="txt" href="#E8:98_2"><span class="lab"><font color="Green" title="E72">A26</font></span></a>, <a class="txt" href="#E6:98_2"><span class="lab"><font color="Green" title="E71">A25</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:98_2"/><span class="lab"><font color="Green" title="E75">A29</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">k1</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">consider </span><font color="Maroon" title="c10">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:98_2"/><span class="lab"><font color="Green" title="E76">A30</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_2"><span class="lab"><font color="Green" title="E69">A23</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="E18:98_2"/><span class="lab"><font color="Green" title="E77">A31</font></span>: 
<font color="Maroon" title="c10">m1</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="txt" href="#E11:98"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E76">A30</font></span></a></span>;<br/>
<a NAME="E19:98_2"/><span class="lab"><font color="Green" title="E78">A32</font></span>: 
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:98_2_2"/>
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:98_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">m1</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:98_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_2"><span class="lab"><font color="Green" title="E70">A24</font></span></a>, <a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E76">A30</font></span></a>, <a class="txt" href="#E18:98_2"><span class="lab"><font color="Green" title="E77">A31</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</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>
<a NAME="E20:98_2"/>
<font color="Maroon" title="c10">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E76">A30</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="E21:98_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/>
<a NAME="E22:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A33</font></span>: 
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98_2"><span class="lab"><font color="Green" title="E78">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E23:98_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">k1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_2"><span class="lab"><font color="Green" title="E74">A28</font></span></a>, <a class="txt" href="#E15:98_2"><span class="lab"><font color="Green" title="E75">A29</font></span></a>, <a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E76">A30</font></span></a></span>;<br/>
<a NAME="E24:98_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98_2"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/>
<a NAME="E25:98_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A34</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</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="E26:98_2"/>
 not <font color="Maroon" title="c4">p9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E17:98_2"><span class="lab"><font color="Green" title="E76">A30</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:98_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="txt" href="#E22:98_2"><span class="lab"><font color="Green" title="E79">A33</font></span></a>, <a class="txt" href="#E25:98_2"><span class="lab"><font color="Green" title="E80">A34</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>
<a NAME="E20:98"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:98_3"/><span class="lab"><font color="Green" title="E68">A35</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:98_3"/><span class="lab"><font color="Green" title="E69">A36</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:98_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E4:98_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_3"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="txt" href="#E1:98_3"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:98_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:98"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E65">A13</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="E21:98"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E67">A21</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>
