<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">n0</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n0</font> &amp; <font color="Maroon" title="c2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n0</font> &amp; <font color="Olive" title="b1">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Olive" title="b1">n0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n0</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</font> &amp; <font color="Maroon" title="c2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</font> &amp; <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</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="c2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</font> or  not <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">m</font> or  not <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:48"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</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="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">m</font> or  not <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:48"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">m</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="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:48"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</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">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:48"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or  not <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:48"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:48"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_1"/>
( <font color="Maroon" title="c3">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c3">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="wsierp_1.html#T15" title="WSIERP_1:th.15">WSIERP_1:15</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="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<font color="Maroon" title="c3">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">X2</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n0</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">X1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font></span><br/><span class="kw">assume </span><a NAME="E1:48_1_2_1"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</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="E2:48_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:48_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E3:48_1_2_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</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="E4:48_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">X5</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T4" target="_self" title="NAT_5:th.4">Th4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">X4</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T4" target="_self" title="NAT_5:th.4">Th4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">X6</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T4" target="_self" title="NAT_5:th.4">Th4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">X3</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T5" target="_self" title="NAT_5:th.5">Th5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">X</font> = <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">1,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n0</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="nat_5.html#E8"><span class="lab"><font color="Green" title="E8">Lm3</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Y</font> = <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X</font>;<br/><a NAME="E11:48_1_2"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">X6</font></span><br/><span class="kw">assume </span><a NAME="E1:48_1_2_2"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">X6</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="E2:48_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:48_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X6</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_2"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E3:48_1_2_2"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E13:48_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">X5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">X6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E14:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
( <font color="Maroon" title="c9">X3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X5</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">X6</font> &amp; <font color="Maroon" title="c6">X5</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X6</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E15:48_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_2_3"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font>
 ; <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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x9</font> = <font color="Maroon" title="c12">x</font> as    <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>  ;<br/>
<a NAME="E3:48_1_2_3"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_3"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/>
<a NAME="E4:48_1_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">x9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="int_2.html#T3" title="INT_2:th.3">INT_2:3</a>, <a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:48_1_2_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font>
 ; <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="E16:48_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E17:48_1_2"/><span class="kw">then </span>
 <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A17</font></span>:  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font> = 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D2" target="_self" title="NAT_5:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n0</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">X</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">X4</font></span><br/><span class="kw">assume </span><a NAME="E1:48_1_2_5"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">X4</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="E2:48_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:48_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A19</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_2_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_5"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E3:48_1_2_5"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 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><a NAME="E20:48_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">X3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">X4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E21:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A20</font></span>: 
( <font color="Maroon" title="c4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X3</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">X4</font> &amp; <font color="Maroon" title="c9">X3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">X4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" title="ENUMSET1:th.3">ENUMSET1:3</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E22:48_1_2"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T4" title="ENUMSET1:th.4">ENUMSET1:4</a></span>;<br/><span class="kw">then </span> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c10">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_2"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T27" target="_self" title="NAT_5:th.27">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c9">X3</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_2"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c6">X5</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c8">X6</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c8">X6</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T27" target="_self" title="NAT_5:th.27">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c8">X6</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T27" target="_self" title="NAT_5:th.27">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c7">X4</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T27" target="_self" title="NAT_5:th.27">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D1" target="_self" title="NAT_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D1" target="_self" title="NAT_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D1" target="_self" title="NAT_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n0</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D1" target="_self" title="NAT_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n0</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n0</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n0</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E24:48_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c3">n0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E11:48_1_2"><span class="lab"><font color="Green" title="E53">A12</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><span class="kw">end;</span></div>

</div>
