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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<a NAME="E1:45"/>
( not <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> implies  not 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E32">A1</font></i>: 
not <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E2:45_1"/>
not 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_1_1"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="pepin.html#E1:45_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="pepin.html#E44"><i><font color="Green" title="E30">Lm13</font></i></a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/>
<a NAME="E3:45_1_1"/><b>then </b>
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:45_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1"/>
not 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
( 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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