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

<span class="kw">let </span><font color="Maroon" title="c1">o</font> be   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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  <a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">R</font> =  <a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">r</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">ir</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">cr</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #);<br/>
<span class="kw">assume </span><a NAME="E1:36"/>
 <a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</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/>

<a NAME="E2:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> 
 
;<br/>
<a NAME="E3:36"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T15" onmouseover="rs('orders_1/T15')" onmouseout="rh()">ORDERS_1:15</a></span>;<br/>
<a NAME="E4:36"/><span class="kw">then </span>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="wellord1.html#T3" onmouseover="rs('wellord1/T3')" onmouseout="rh()">WELLORD1:3</a></span>;<br/>
<a NAME="E5:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 
;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> ($1,1);<br/>
<div><span class="lab"><font color="Green" title="E21">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </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">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">set </span><font color="Maroon" title="c7">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<font color="Maroon" title="c6">n</font>,1);<br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<font color="Maroon" title="c6">n</font>,1) as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">o</font> ;<br/><a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T85" onmouseover="rs('card_3/T85')" onmouseout="rh()">CARD_3:85</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; ( <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Olive" title="b1">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( ( <font color="Maroon" title="c9">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; ( <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c9">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><br/><div><span class="kw">hereby </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"> ( <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c9">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c9">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"> <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><a NAME="E2:36_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</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><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:36_1_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:36_1_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 not <font color="Maroon" title="c9">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"><font color="Maroon" title="c6">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"> contradiction</span><br/><a NAME="E4:36_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:36_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><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="E1:36_1_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> or  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:36_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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="36_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_2"/>
 not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:36_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E5:36_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</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><span class="kw">end;</span></div><a NAME="E5:36_1"/><span class="kw">then </span>
 <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <font color="Maroon" title="c8">y</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"><font color="Maroon" title="c6">n</font></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="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/><a NAME="E6:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> is  <a href="pre_poly.html#V2" title="PRE_POLY:attr.2">finite-support</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D8" onmouseover="rs('pre_poly/D8')" onmouseout="rh()">PRE_POLY:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" onmouseover="rs('pre_poly/D12')" onmouseout="rh()">PRE_POLY:def 12</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font>; <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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c10">y</font>]</span><br/><span class="kw">thus </span><a NAME="E8:36_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c10">y</font>]
 ; <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>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)<span class="kw"> such that </span><br/><a NAME="E8:36"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c6">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E21">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) ;<br/>
<a NAME="E10:36"/>
<font color="Maroon" title="c7">f</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellfnd1.html#D6" onmouseover="rs('wellfnd1/D6')" onmouseout="rh()">WELLFND1:def 6</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 <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">n0</font> = <font color="Maroon" title="c8">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">fn1</font> = <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">fn</font> = <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n0</font>;<br/>
<a NAME="E2:36_2"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E22">A9</font></span></a></span>;<br/>
<a NAME="E3:36_2"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<font color="Maroon" title="c8">n</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36"><span class="lab"><font color="Green" title="E22">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">fn1</font> = <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">fn</font> = <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n0</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/>
<a NAME="E6:36_2"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c9">n0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
;<br/>
<a NAME="E7:36_2"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T85" onmouseover="rs('card_3/T85')" onmouseout="rh()">CARD_3:85</a></span>;<br/>
<a NAME="E8:36_2"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: <font color="Maroon" title="c12">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E7:36_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
;<br/>
<a NAME="E10:36_2"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E11:36_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c13">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E7:36_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_2_2"><span class="kw">now </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">  for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">n</font> holds <br/><font color="Maroon" title="c12">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">l</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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"> ( <font color="Maroon" title="c14">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">n</font> implies <font color="Maroon" title="c12">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:36_2_2"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c14">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">n</font>
 ; <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"> <font color="Maroon" title="c12">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font></span><br/><a NAME="E2:36_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c14">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">n</font>
 ;<br/><a NAME="E3:36_2_2"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:36_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span> } </span> 
 ;<br/><a NAME="E5:36_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="axioms.html#T4" onmouseover="rs('axioms/T4')" onmouseout="rh()">AXIOMS:4</a></span>;<br/><a NAME="E6:36_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/><a NAME="E7:36_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_2_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a></span>;<br/><a NAME="E8:36_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c12">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">o</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>
<br/>.= 
<font color="Maroon" title="c13">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E2:36_2_2"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>
;<br/> <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>
<a NAME="E13:36_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<font color="Maroon" title="c12">fn1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c13">fn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E11:36_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="pre_poly.html#D9" onmouseover="rs('pre_poly/D9')" onmouseout="rh()">PRE_POLY:def 9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:36_2"/>
<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E7:36_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E9:36_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E10:36_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)</span><br/>

<a NAME="E15:36_2"/>
<font color="Maroon" title="c12">fn1</font> <a href="pre_poly.html#R2" title="PRE_POLY:pred.2">&lt;='</a> <font color="Maroon" title="c13">fn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_2"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="ref" href="pre_poly.html#D10" onmouseover="rs('pre_poly/D10')" onmouseout="rh()">PRE_POLY:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:36_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#NK4" title="BAGORDER:NK.4">LexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D14" onmouseover="rs('pre_poly/D14')" onmouseout="rh()">PRE_POLY:def 14</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>
<span class="kw">hence </span><a NAME="E11:36"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="wellfnd1.html#T14" onmouseover="rs('wellfnd1/T14')" onmouseout="rh()">WELLFND1:14</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>
