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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <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> ; <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">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> implies ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) ) &amp; ( ( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) implies ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> implies ( (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) ) )</span><br/>



<a NAME="E1:59"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:59"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> implies ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <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="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) implies ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> implies ( (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) ) )</span><br/>

<a NAME="E3:59"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> implies ( (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_1_1_1"/>
(  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )</span><br/><span class="kw">thus </span><a NAME="E3:59_1_1_1"/>
(  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_1"><span class="lab"><font color="Green" title="E51">A4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_1_2"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp; (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_1_1_2"/>
(  not 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )</span><br/><span class="kw">thus </span><a NAME="E3:59_1_1_2"/>
(  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E1:59_1_1_2"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">and </span><br/><a NAME="E6:59"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> ) &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:59"/>
(  not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E50">A6</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"> (  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:59"/>
(  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp;  not <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</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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