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

<span class="kw">set </span><font color="Maroon" title="c2">IR</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">CR</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">FIR</font> =  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">FCR</font> =  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E1:67_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font></span>)</span> #)
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_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>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></span>)</span>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> or <font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font></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"> ( <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y9</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> );<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_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">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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"> (  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><span class="kw">assume </span><a NAME="E1:67_1_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><a NAME="E2:67_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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><a NAME="E3:67_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) ) holds <br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) ) implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:67_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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"> (  not  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><span class="kw">assume </span><a NAME="E2:67_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1_2_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1_1_2_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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="67_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_2_1_2_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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1_2_1_2_1_1"/>
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1_1_2_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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="67_1_1_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1"/>
(  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font> or  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1"/>
(  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font> or  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D29" onmouseover="rs('waybel_0/D29')" onmouseout="rh()">WAYBEL_0:def 29</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><a NAME="E2:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</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 <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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="67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/>
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><a NAME="E2:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</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 <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:67_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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><span class="kw">hence </span><a NAME="E3:67_1_1_1_2_1_2_1_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">ax</font> = <font color="Maroon" title="c11">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">bx</font> = <font color="Maroon" title="c12">b</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E2:67_1_1_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1_1_2"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="bagorder.html#E52"><span class="lab"><font color="Green" title="E35">Lm1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">ax</font> = <font color="Maroon" title="c11">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c16">bx</font> = <font color="Maroon" title="c12">b</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T37" target="_self" onmouseover="rs('bagorder/T37')" onmouseout="rh()">Th34</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_2_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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_2_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">ax</font>,<font color="Maroon" title="c16">bx</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">bx</font>,<font color="Maroon" title="c15">ax</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1_2"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E2:67_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">ax</font>,<font color="Maroon" title="c16">bx</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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="67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">bx</font>,<font color="Maroon" title="c15">ax</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:67_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="bagorder.html#T39" target="_self" onmouseover="rs('bagorder/T39')" onmouseout="rh()">Th36</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><span class="kw">hence </span><a NAME="E5:67_1_1_1_2_1_2_1_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:67_1_1_1_2_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:67_1_1_1_2_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c11">a</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:67_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/><a NAME="E6:67_1_1_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E3:67_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>);<br/></span><a NAME="E7:67_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">x</font>
 ;<br/><a NAME="E8:67_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x9</font>,<font color="Maroon" title="c9">y9</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y9</font>,<font color="Maroon" title="c8">x9</font></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>  <a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:67_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</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="E3:67_1_1"/>
 <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <font color="Maroon" title="c1">R</font> is  <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D29" onmouseover="rs('waybel_0/D29')" onmouseout="rh()">WAYBEL_0:def 29</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>
