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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="ordinal6.html#V1" title="ORDINAL6:attr.1">ordinal-membered</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> ) implies <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E19">Z0</font></span>: 
 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>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">g</font> =  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">a</font> =  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b</font> =  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Q</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R1</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Q1</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">R2</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Q2</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c2">Y</font>;<br/>
<a NAME="E2:46"/><span class="lab"><font color="Green" title="E20">X</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></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> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/>
<a NAME="E3:46"/><span class="lab"><font color="Green" title="E21">A0</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T18" target="_self" title="ORDINAL6:th.18">ThN1</a></span>;<br/>
<a NAME="E4:46"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T19" target="_self" title="ORDINAL6:th.19">ThN1a</a></span>;<br/>
<a NAME="E5:46"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
(  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font> <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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font> <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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T21" target="_self" title="ORDINAL6:th.21">ThN3</a></span>;<br/>
<a NAME="E6:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
(  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font> 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/>
<span class="kw">thus </span><span class="lab"><font color="Green" title="E25">A5</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>
<br/>.= 
 <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</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#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ) ) ) )</span><br/>

<span class="kw">thus </span> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> = 
<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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T10" target="_self" title="ORDINAL6:th.10">ORDINAL4th2</a></span>
<br/>.= 
 <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"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ) ) ) )</span><br/>

<a NAME="E9:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</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="E10:46"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
(  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T2" target="_self" title="ORDINAL6:th.2">EXCH10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:46"/>
<span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c13">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or  not <font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c14">y</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:46_3"/><span class="lab"><font color="Green" title="E28">B1</font></span>: 
( <font color="Maroon" title="c13">x</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"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_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="E1:46_3_1"/>
( ( <font color="Maroon" title="c13">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or ( <font color="Maroon" title="c13">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</font> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_3_1_1"/><span class="lab"><font color="Green" title="E29">B2</font></span>: 
( <font color="Maroon" title="c13">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font></span><br/><div class="add"><a NAME="E2:46_3_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_3_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="txt" href="#E1:46_3_1_1"><span class="lab"><font color="Green" title="E29">B2</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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="46_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_3_1_2"/><span class="lab"><font color="Green" title="E29">B3</font></span>: 
( <font color="Maroon" title="c13">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font></span><br/><div class="add"><a NAME="E2:46_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B4</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B5</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E20">X</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_3_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3_1_2"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="txt" href="#E2:46_3_1_2"><span class="lab"><font color="Green" title="E30">B4</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E5:46_3_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="txt" href="#E1:46_3_1_2"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="txt" href="#E3:46_3_1_2"><span class="lab"><font color="Green" title="E31">B5</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="E6:46_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E19">Z0</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_3_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:46_3_1_3"/><span class="lab"><font color="Green" title="E29">B3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font></span><br/><div class="add"><a NAME="E2:46_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B4</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B5</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E20">X</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_3_1_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3_1_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="txt" href="#E2:46_3_1_3"><span class="lab"><font color="Green" title="E30">B4</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E5:46_3_1_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="txt" href="#E1:46_3_1_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="txt" href="#E3:46_3_1_3"><span class="lab"><font color="Green" title="E31">B5</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="E6:46_3_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E19">Z0</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_3_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:46_3_1_4"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font></span><br/><div class="add"><a NAME="E2:46_3_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">B4</font></span>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_3_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B5</font></span>: 
( <font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E20">X</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_3_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_1_4"><span class="lab"><font color="Green" title="E29">B4</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_3_1_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="txt" href="#E2:46_3_1_4"><span class="lab"><font color="Green" title="E29">B4</font></span></a>, <a class="txt" href="#E3:46_3_1_4"><span class="lab"><font color="Green" title="E30">B5</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) ) &amp; (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ) )</span><br/>



<a NAME="E12:46"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</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/>
<div><span class="kw">hereby </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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> or  not <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_4"/><span class="lab"><font color="Green" title="E29">C1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/><span class="kw">hence </span><a NAME="E2:46_4"/><span class="lab"><font color="Green" title="E30">C2</font></span>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><a NAME="E3:46_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">C3</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_4"><span class="lab"><font color="Green" title="E29">C1</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E4:46_4"/><span class="lab"><font color="Green" title="E32">C4</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E2:46_4"><span class="lab"><font color="Green" title="E30">C2</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">thus </span><a NAME="E5:46_4"/>
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c14">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="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E2:46_4"><span class="lab"><font color="Green" title="E30">C2</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_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="E1:46_4_1_1"/>
( ( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or ( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_4_1_1_1"/><span class="lab"><font color="Green" title="E33">B2</font></span>: 
( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><div class="add"><a NAME="E2:46_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E3:46_4"><span class="lab"><font color="Green" title="E31">C3</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E3:46_4_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_4_1_1_1"><span class="lab"><font color="Green" title="E33">B2</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E4:46_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">B7</font></span>: 
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E2:46_4_1_1_1"><span class="lab"><font color="Green" title="E34">B6</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/><a NAME="E5:46_4_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">X</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E6:46_4_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="E7:46_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_4_1_1_1"><span class="lab"><font color="Green" title="E35">B7</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:46_4_1_1_1"/>
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_4"><span class="lab"><font color="Green" title="E32">C4</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_4_1_1_2"/><span class="lab"><font color="Green" title="E33">B3</font></span>: 
( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><div class="add"><a NAME="E2:46_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B4</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">B5</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E2:46_4"><span class="lab"><font color="Green" title="E30">C2</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_4_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_4_1_1_2"><span class="lab"><font color="Green" title="E33">B3</font></span></a>, <a class="txt" href="#E2:46_4_1_1_2"><span class="lab"><font color="Green" title="E34">B4</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E5:46_4_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:46_4_1_1_2"><span class="lab"><font color="Green" title="E33">B3</font></span></a>, <a class="txt" href="#E3:46_4_1_1_2"><span class="lab"><font color="Green" title="E35">B5</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="E6:46_4_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E19">Z0</font></span></a></span>;<br/><a NAME="E7:46_4_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:46_4_1_1_2"/>
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_4"><span class="lab"><font color="Green" title="E32">C4</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:46_4_1_1_3"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><div class="add"><a NAME="E2:46_4_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x</font>
 ;<br/><a NAME="E3:46_4_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_4"><span class="lab"><font color="Green" title="E31">C3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:46_4_1_1_3"/>
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:46_4_1_1_4"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/><div class="add"><a NAME="E2:46_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B4</font></span>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B5</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E2:46_4"><span class="lab"><font color="Green" title="E30">C2</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_4_1_1_4"/>
<font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_4"><span class="lab"><font color="Green" title="E31">C3</font></span></a>, <a class="ref" href="ordinal3.html#T59" title="ORDINAL3:th.59">ORDINAL3:59</a></span>;<br/><a NAME="E5:46_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">B6</font></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="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span></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> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_4_1_1_4"><span class="lab"><font color="Green" title="E34">B5</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E6:46_4_1_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_4_1_1_4"><span class="lab"><font color="Green" title="E33">B4</font></span></a>, <a class="txt" href="#E3:46_4_1_1_4"><span class="lab"><font color="Green" title="E34">B5</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E7:46_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">B7</font></span>: 
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:46_4_1_1_4"><span class="lab"><font color="Green" title="E35">B6</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/><a NAME="E8:46_4_1_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">Y</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E9:46_4_1_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</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="E10:46_4_1_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_4_1_1_4"><span class="lab"><font color="Green" title="E36">B7</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:46_4_1_1_4"/>
<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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_4"><span class="lab"><font color="Green" title="E32">C4</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

<span class="kw">assume </span><a NAME="E14:46"/><span class="lab"><font color="Green" title="E29">D1</font></span>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/>

<a NAME="E15:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">D2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T8" target="_self" title="ORDINAL6:th.8">Lem3</a></span>;<br/>
<a NAME="E16:46"/><span class="lab"><font color="Green" title="E31">D3</font></span>: 
(  <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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c14">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="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5"><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="E1:46_5"/>
( ( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or ( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> ) or (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_1"><span class="kw">suppose </span></a><a NAME="E1:46_5_1"/><span class="lab"><font color="Green" title="E32">B2</font></span>: 
( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:46_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B3</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E3:46_5_1"/>
( <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:46_5_1"><span class="lab"><font color="Green" title="E32">B2</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="E4:46_5_1"/><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"><span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_1"><span class="lab"><font color="Green" title="E33">B3</font></span></a>, <a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E30">D2</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E5:46_5_1"/><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="c15">x</font>,<font color="Maroon" title="c16">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="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E1:46_5_1"><span class="lab"><font color="Green" title="E32">B2</font></span></a>, <a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E31">D3</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/><a NAME="E6:46_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T8" target="_self" title="ORDINAL6:th.8">Lem3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_5_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_2"><span class="kw">suppose </span></a><a NAME="E1:46_5_2"/>
( <font color="Maroon" title="c15">x</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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:46_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_5_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3"><span class="kw">suppose </span></a><a NAME="E1:46_5_3"/><span class="lab"><font color="Green" title="E32">B2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:46_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E3:46_5_3"/><span class="lab"><font color="Green" title="E34">B4</font></span>: 
<span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:46_5_3"><span class="lab"><font color="Green" title="E32">B2</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="E4:46_5_3"/><span class="lab"><font color="Green" title="E35">B7</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_5_3"><span class="lab"><font color="Green" title="E32">B2</font></span></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E5:46_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">B5</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E6:46_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">B6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_5_3"><span class="lab"><font color="Green" title="E35">B7</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E7:46_5_3"/>
<span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E5:46_5_3"><span class="lab"><font color="Green" title="E36">B5</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="E8:46_5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_3"><span class="lab"><font color="Green" title="E33">B3</font></span></a>, <a class="txt" href="#E3:46_5_3"><span class="lab"><font color="Green" title="E34">B4</font></span></a>, <a class="txt" href="#E6:46_5_3"><span class="lab"><font color="Green" title="E37">B6</font></span></a>, <a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E19">Z0</font></span></a></span>;<br/><a NAME="E9:46_5_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E30">D2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:46_5_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</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="46_5_4"><span class="kw">suppose </span></a><a NAME="E1:46_5_4"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:46_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">B4</font></span>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></span>;<br/><a NAME="E3:46_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B5</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <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"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="ordinal3.html#T22" title="ORDINAL3:th.22">ORDINAL3:22</a></span>;<br/><a NAME="E4:46_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B6</font></span>: 
( <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E22">A1</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="E5:46_5_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_4"><span class="lab"><font color="Green" title="E32">B4</font></span></a>, <a class="txt" href="#E3:46_5_4"><span class="lab"><font color="Green" title="E33">B5</font></span></a>, <a class="ref" href="ordinal4.html#D1" title="ORDINAL4:def.1">ORDINAL4:def 1</a></span>;<br/><a NAME="E6:46_5_4"/><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"><span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span></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="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_5_4"><span class="lab"><font color="Green" title="E34">B6</font></span></a>, <a class="txt" href="#E15:46"><span class="lab"><font color="Green" title="E30">D2</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E7:46_5_4"/><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="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></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> <span class="p1">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E31">D3</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:46_5_4"><span class="lab"><font color="Green" title="E33">B5</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/><a NAME="E8:46_5_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K2" title="ORDINAL6:func.2">numbering</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T8" target="_self" title="ORDINAL6:th.8">Lem3</a></span>;<br/><a NAME="E9:46_5_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_4"><span class="lab"><font color="Green" title="E32">B4</font></span></a>, <a class="ref" href="ordinal3.html#T18" title="ORDINAL3:th.18">ORDINAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E10:46_5_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">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"><span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal6.html#K1" title="ORDINAL6:func.1">ord-type</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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