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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a> <font color="Olive" title="b1">B</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">M</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E1:9"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</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="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:9"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> $1</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="ordinal2.html#D4" title="ORDINAL2:def.4">ORDINAL2:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:9"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">B</font> =  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>;<br/>
<span class="kw">take </span>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> )</span><br/>

<a NAME="E10:9"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </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">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:9_1"/>
(  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:9"/>
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T14" title="WELLORD2:th.14">WELLORD2:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">phi</font> =  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>);<br/>
<a NAME="E12:9"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="wellord2.html#T8" title="WELLORD2:th.8">WELLORD2:8</a></span>;<br/>
<a NAME="E13:9"/><span class="kw">then </span>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D2" title="WELLORD2:def.2">WELLORD2:def 2</a></span>;<br/>
<a NAME="E14:9"/><span class="kw">then </span>
(  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T40" title="WELLORD1:th.40">WELLORD1:40</a>, <a class="ref" href="wellord2.html#T8" title="WELLORD2:th.8">WELLORD2:8</a></span>;<br/>
<a NAME="E15:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>) <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D9" title="WELLORD1:def.9">WELLORD1:def 9</a></span>;<br/>
<a NAME="E16:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E17:9"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E18:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E19:9"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E20:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">phi</font> =  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> (<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>) as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">phi</font> = <font color="Maroon" title="c10">phi</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="ordinal2.html#D4" title="ORDINAL2:def.4">ORDINAL2:def 4</a></span>;<br/>
<a NAME="E23:9"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">xi</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">phi</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></span>;<br/>
<a NAME="E25:9"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">phi</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">xi</font> = <font color="Maroon" title="c12">xi</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D4" title="ORDINAL2:def.4">ORDINAL2:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">xi</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c13">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c13">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">xi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font> )</span><br/>

<span class="kw">thus </span><a NAME="E27:9"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">xi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font> )</span><br/>

<span class="kw">thus </span><a NAME="E28:9"/>
<font color="Maroon" title="c13">xi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> or  not <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c13">xi</font> or <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_2"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">and </span><br/><a NAME="E2:9_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">xi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font></span><br/>

<a NAME="E3:9_2"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E4:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:9_2"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a></span>;<br/>
<a NAME="E6:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E16:9"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E3:9_2"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">a9</font> = <font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c17">b9</font> = <font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">a99</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a9</font>, <font color="Maroon" title="c19">b99</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b9</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E9:9_2"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:9_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">b99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">b9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>;<br/>
<a NAME="E11:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A17</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">b9</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c19">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/>
<a NAME="E12:9_2"/>
<font color="Maroon" title="c14">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E13:9_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E3:9_2"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E14:9_2"/><span class="kw">then </span>
<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="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</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>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<a NAME="E15:9_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_2"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E9:9_2"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E16:9_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_2"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E17:9_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a></span>;<br/>
<a NAME="E18:9_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">a99</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">b9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E4:9_2"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="funct_1.html#T50" title="FUNCT_1:th.50">FUNCT_1:50</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E19:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c18">a99</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c19">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_2"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:9_2"/>
<font color="Maroon" title="c18">a99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E6:9_2"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E4:9_2"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E9:9_2"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E21:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c18">a99</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c19">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:9_2"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E22:9_2"/>
( <font color="Maroon" title="c18">a99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c19">b99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:9_2"/>
<font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:9_2"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E29:9"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:9_3"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:9_3"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c15">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E4:9_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_3_1_1"/>
(  not <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_3_1_1_1"/>
 not <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font></span><br/><div class="add"><a NAME="E2:9_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E21">A20</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:9_3_1_1_1"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:9_3_1_1_1"/>
( <font color="Maroon" title="c16">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> &amp; <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">xi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:9_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E4:9_3_1_1_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:9_3_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_3_1_1_2"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><span class="kw">consider </span><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:9_3_1_1_2"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:9_3_1_1_2"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_3_1_1_2"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="card_5.html#T2" target="_self" title="CARD_5:th.2">Th10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">z</font> = <font color="Maroon" title="c18">z</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3_1_1_2"><span class="lab"><font color="Green" title="E24">A24</font></span></a></span>;<br/><a NAME="E6:9_3_1_1_2"/><span class="lab"><font color="Green" title="E26">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3_1_1_2"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E7:9_3_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E8:9_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_3_1_1_2"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:9_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A27</font></span>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_3_1_1_2"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="txt" href="#E6:9_3_1_1_2"><span class="lab"><font color="Green" title="E26">A26</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E11:9_3_1_1_2"/><span class="lab"><font color="Green" title="E28">A28</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">C</font>] &amp; (  for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c20">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E9:9_3_1_1_2"><span class="lab"><font color="Green" title="E27">A27</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:9_3_1_1_2_1"/>
<font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E11:9_3_1_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span></span><br/><span class="kw">assume </span><a NAME="E2:9_3_1_1_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:9_3_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A29</font></span>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:9_3_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T2" target="_self" title="CARD_5:th.2">Th10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c21">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E29">A29</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">fa</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>, <font color="Maroon" title="c24">fy</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E8:9_3_1_1_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">fa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E29">A29</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E9:9_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A31</font></span>: 
<font color="Maroon" title="c24">fy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_3_1_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E5:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E30">A30</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><a NAME="E10:9_3_1_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E11:9_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A32</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E29">A29</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E12:9_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_3_1_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E13:9_3_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_3_1_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E11:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E32">A32</font></span></a>, <a class="txt" href="#E9:9_3_1_1_2_1"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:9_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:9_3_1_1_2"/><span class="lab"><font color="Green" title="E29">A33</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c6">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:9_3_1_1_2"/><span class="lab"><font color="Green" title="E30">A34</font></span>: 
<font color="Maroon" title="c20">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">z</font> = <font color="Maroon" title="c21">z</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9_3_1_1_2"><span class="lab"><font color="Green" title="E29">A33</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">xz</font> = <font color="Maroon" title="c13">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">z</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E19:9_3_1_1_2"/>
<font color="Maroon" title="c23">xz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">xi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E15:9_3_1_1_2"><span class="lab"><font color="Green" title="E29">A33</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E20:9_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A35</font></span>: 
<font color="Maroon" title="c23">xz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/><a NAME="E21:9_3_1_1_2"/>
<font color="Maroon" title="c16">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">xz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:9_3"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E11:9_3_1_1_2"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E15:9_3_1_1_2"><span class="lab"><font color="Green" title="E29">A33</font></span></a>, <a class="txt" href="#E16:9_3_1_1_2"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E22:9_3_1_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:9_3_1_1_2"><span class="lab"><font color="Green" title="E31">A35</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:9_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E30:9"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T18" title="ORDINAL2:th.18">ORDINAL2:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:9"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c13">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="ordinal2.html#T22" title="ORDINAL2:th.22">ORDINAL2:22</a>, <a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
