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

<span class="kw">let </span><font color="Maroon" title="c1">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"> (  not 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> iff  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">K</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font> or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T29" target="_self" onmouseover="rs('number02/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:38"/>
(  not 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E34">A1</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"> (  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 ) implies  not 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E5:38"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 )
 ; <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">  not 9 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font></span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">t</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E7:38"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_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="E1:38_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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/><div class="add"><a NAME="E2:38_1_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <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/><div class="add"><a NAME="E2:38_1_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_3"><span class="kw">suppose </span></a><a NAME="E1:38_1_3"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 ; <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/><div class="add"><a NAME="E2:38_1_3"/><span class="kw">then </span>
3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_4"><span class="kw">suppose </span></a><a NAME="E1:38_1_4"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 ; <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/><div class="add"><a NAME="E2:38_1_4"/><span class="kw">then </span>
4 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_5"><span class="kw">suppose </span></a><a NAME="E1:38_1_5"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5
 ; <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/><div class="add"><a NAME="E2:38_1_5"/><span class="kw">then </span>
5 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_6"><span class="kw">suppose </span></a><a NAME="E1:38_1_6"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6
 ; <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/><div class="add"><a NAME="E2:38_1_6"/><span class="kw">then </span>
6 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_7"><span class="kw">suppose </span></a><a NAME="E1:38_1_7"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7
 ; <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/><div class="add"><a NAME="E2:38_1_7"/><span class="kw">then </span>
7 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="38_1_8"><span class="kw">suppose </span></a><a NAME="E1:38_1_8"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">9 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8
 ; <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/><div class="add"><a NAME="E2:38_1_8"/><span class="kw">then </span>
8 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 9 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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>
