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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>set </b><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="E1:11"/>
 <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> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:11"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <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> &amp;  <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> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>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/>
<b>consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:11"/><i><font color="Green" title="E11">A2</font></i>: 
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="c4">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>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font>;<br/>
<b>set </b><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="c4">X</font></span>)</span>;<br/>
<b>set </b><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="c4">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="c4">X</font></span>)</span>;<br/>
<b>take </b>
 <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="c4">X</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">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="c4">X</font></span>)</span> )</span><br/>

<a NAME="E8:11"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">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="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c10">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/>

<b>thus </b><a NAME="E1:11_1"/>
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c10">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> )
 <b>by </b><i><a class="txt" href="card_5.html#E5:11"><i><font color="Green" title="E11">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:11"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
(  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font> 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="c4">X</font></span>)</span></span>)</span> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T9" title="WELLORD2:th.9">WELLORD2:9</a></i>;<br/>
<a NAME="E10:11"/><b>then </b>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">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="c4">X</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D2" title="WELLORD2:def.2">WELLORD2:def 2</a></i>;<br/>
<a NAME="E11:11"/><b>then </b>
 <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="c4">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></i>;<br/>
<a NAME="E12:11"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
(  <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="c4">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="c4">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="c4">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font> &amp;  <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="c4">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="c4">X</font></span>)</span> &amp;  <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="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E9:11"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="wellord1.html#D9" title="WELLORD1:def.9">WELLORD1:def 9</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E13:11"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
(  <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="c4">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="c4">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="c4">X</font></span>)</span> &amp;  <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="c4">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="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> &amp;  <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="c4">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="c4">X</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c4">X</font></span>)</span></span>)</span> iff ( <font color="Olive" title="b1">x</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="c4">X</font></span>)</span> &amp; <font color="Olive" title="b2">y</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="c4">X</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p0">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p0">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</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="c4">X</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<b>then reconsider </b><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="c4">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="c4">X</font></span>)</span> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<b>thus </b><a NAME="E16:11"/>
 <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="c4">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>
 <b>by </b><i><a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="wellord2.html#T17" title="WELLORD2:th.17">WELLORD2:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">X</font></span>)</span></span><br/>

<a NAME="E17:11"/><i><font color="Green" title="E16">A7</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">X</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> )
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">xi</font> = <font color="Maroon" title="c6">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> <b>by </b><i><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="card_5.html#E17:11"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">xi</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D21" title="ORDINAL2:def.21">ORDINAL2:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c4">X</font></span>)</span> &amp;  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">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> &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/>

<b>thus </b><a NAME="E20:11"/>
(  <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="c4">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> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E21:11"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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="11_2"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#NK2" title="RELAT_1:NK.2">dom</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/>

<b>assume </b><a NAME="E1:11_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <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> &amp; <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:11_2"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <font color="Maroon" title="c13">xi</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:11_2"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
( <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> &amp; <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="c4">X</font></span>)</span></span>)</span> &amp; <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="c4">X</font> &amp; <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="c4">X</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="card_5.html#E17:11"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="card_5.html#E1:11_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a'</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">b'</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/>
<b>reconsider </b><font color="Maroon" title="c18">a''</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">a'</font>, <font color="Maroon" title="c19">b''</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b'</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E6:11_2"/>
<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="c4">X</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E12:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="card_5.html#E3:11_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E7:11_2"/><b>then </b>
<font color="Maroon" title="c16">a'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">b'</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:11_2"/><b>then </b>
<font color="Maroon" title="c16">a'</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c17">b'</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E9:11_2"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
( <font color="Maroon" title="c16">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">b'</font> &amp; not <font color="Maroon" title="c19">b''</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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span> &amp; <font color="Maroon" title="c18">a''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b''</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E5:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_5.html#E3:11_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E10:11_2"/><b>then </b>
( <font color="Maroon" title="c18">a''</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E8:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_5.html#E3:11_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/>
<a NAME="E11:11_2"/><b>then </b>
( <font color="Maroon" title="c18">a''</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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">b'</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c19">b''</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E9:11_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:11_2"/><b>then </b><i><font color="Green" title="E21">A12</font></i>: 
( <font color="Maroon" title="c18">a''</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c19">b''</font> &amp; <font color="Maroon" title="c18">a''</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">b''</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> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:11_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="card_5.html#E2:11_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:11_2"/><b>then </b>
<font color="Maroon" title="c18">a''</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c19">b''</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E9:11_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E14:11_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>
 <b>by </b><i><a class="txt" href="card_5.html#E12:11_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E22:11"/>
<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>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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="11_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:11_3"/><i><font color="Green" title="E17">A13</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_3"/><i><font color="Green" title="E18">A14</font></i>: 
( <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="c6">f</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x'</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> <b>by </b><i><a class="txt" href="card_5.html#E1:11_3"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="card_5.html#E3:11_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_1"/>
(  not <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">y</font></span>)</span> or <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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="11_3_1_1_1"><b>suppose </b></a><a NAME="E1:11_3_1_1_1"/>
not <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:11_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E5:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="card_5.html#E3:11_3"><i><font color="Green" title="E18">A14</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:11_3_1_1_1"/><i><font color="Green" title="E19">A15</font></i>: 
( <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="c4">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> )
 <b>by </b><i><a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:11_3_1_1_1"/>
( <font color="Maroon" title="c16">x'</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> )
 <b>by </b><i><a class="txt" href="card_5.html#E17:11"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="card_5.html#E3:11_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="card_5.html#E4:11_3_1_1_1"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:11_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>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_2"><b>suppose </b></a><a NAME="E1:11_3_1_1_2"/>
<font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>then consider </b><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_3_1_1_2"/><i><font color="Green" title="E19">A16</font></i>: 
( <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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c17">y</font></span>)</span> &amp; <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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> )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self" title="CARD_5:th.10">Th10</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="card_5.html#E3:11_3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c6">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="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E5:11_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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E6:11_3_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:11_3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:11_3_1_1_2"/><b>then </b><i><font color="Green" title="E20">A17</font></i>: 
 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>]
 <b>by </b><i><a class="txt" href="card_5.html#E3:11_3_1_1_2"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:11_3_1_1_2"/><i><font color="Green" title="E21">A18</font></i>: 
( <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> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="card_5.html#E7:11_3_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:11_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>
 <b>by </b><i><a class="txt" href="card_5.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_5.html#E3:11_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="card_5.html#E9:11_3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c20">C</font></span>)</span></span><br/><b>assume </b><a NAME="E2:11_3_1_1_2_1"/>
<font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c20">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c21">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:11_3_1_1_2_1"/><i><font color="Green" title="E22">A19</font></i>: 
( <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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c20">C</font></span>)</span> &amp; <font color="Maroon" title="c6">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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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> )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self" title="CARD_5:th.10">Th10</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="card_5.html#E4:11_3_1_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">fa</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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">fc</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">C</font>, <font color="Maroon" title="c25">fy</font> = <font color="Maroon" title="c6">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="E7:11_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="c6">f</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</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="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c20">C</font> &amp; <font color="Maroon" title="c6">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> &amp; <font color="Maroon" title="c24">fc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">fa</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E4:11_3_1_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E8:11_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E23">A20</font></i>: 
( <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> &amp; <font color="Maroon" title="c25">fy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:11_3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="card_5.html#E4:11_3_1_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:11_3_1_1_2_1"/><b>then </b>
( <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> &amp; not <font color="Maroon" title="c20">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">a</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:11_3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E10:11_3_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_5.html#E9:11_3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="card_5.html#E8:11_3_1_1_2_1"><i><font color="Green" title="E23">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E11:11_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:11"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:11_3_1_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
( <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="c4">X</font></span>)</span> &amp; <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> )
 <b>by </b><i><a class="txt" href="card_5.html#E13:11"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="card_5.html#E13:11_3_1_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><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="E16:11_3_1_1_2"/>
( <font color="Maroon" title="c23">xz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">C</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="card_5.html#E17:11"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="card_5.html#E13:11_3_1_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:11_3_1_1_2"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">xz</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:11_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="card_5.html#E9:11_3_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/><b>hence </b><a NAME="E18:11_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>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:11_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E23:11"/>
 <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>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<b>hence </b><a NAME="E24:11"/>
 <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>
 <b>by </b><i><a class="txt" href="card_5.html#E17:11"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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