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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:61"/><i><font color="Green">E34</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T20">NAT_1:20</a></i>;<br/>
<a NAME="E2:61"/><i><font color="Green">E52</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
;<br/>
<b>thus </b><a NAME="E3:61"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> implies ( (  not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) &amp; not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_1"/>
(  not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>thus </b><a NAME="E3:61_1_1_1"/>
(  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E2:61"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_2"/>
(  not 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><b>thus </b><a NAME="E3:61_1_1_2"/>
(  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="txt" href="graphsp.html#E2:61"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:61"/><i><font color="Green">E63</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/>

<b>hence </b><a NAME="E6:61"/>
(  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/>

<b>thus </b><a NAME="E7:61"/>
( not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="graphsp.html#T4" target="_self">Th4</a>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>


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