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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b1">R</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  &amp;  not <font color="Olive" title="b1">R</font>,<font color="Olive" title="b1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</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> <font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">a</font></span>)</span>,<font color="Olive" title="b1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c2">R</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  &amp;  not <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  implies  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</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> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c2">R</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  or  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</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> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  ) )</span><br/>

<div><span class="lab"><font color="Green" title="E60">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c3">a</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"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:64_1"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:64_1"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="wellord1.html#D8" target="_self" title="WELLORD1:def.8">Def8</a></span>;<br/><div><span class="lab"><font color="Green" title="E64">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">b</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies ( <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="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</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"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )</span><br/><a NAME="E2:64_1_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"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="wellord1.html#T36" target="_self" title="WELLORD1:th.36">Th45</a></span>;<br/><a NAME="E3:64_1_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"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_1"/>
( <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="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="wellord1.html#T36" target="_self" title="WELLORD1:th.36">Th45</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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><a NAME="E7:64_1"/><span class="lab"><font color="Green" title="E65">A9</font></span>: 
<font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="wellord1.html#D4" target="_self" title="WELLORD1:def.4">Def4</a></span>;<br/><a NAME="E8:64_1"/><span class="lab"><font color="Green" title="E66">A10</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="wellord1.html#T31" target="_self" title="WELLORD1:th.31">Th39</a></span>;<br/><a NAME="E9:64_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="wellord1.html#T25" target="_self" title="WELLORD1:th.25">Th32</a>, <a class="ref" href="wellord1.html#T32" target="_self" title="WELLORD1:th.32">Th40</a></span>;<br/><a NAME="E10:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A11</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/><a NAME="E11:64_1"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/><a NAME="E12:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E8:64_1"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E13:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A14</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="wellord1.html#T1" target="_self" title="WELLORD1:th.1">Th1</a></span>;<br/><a NAME="E14:64_1"/>
<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="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="ref" href="wellord1.html#T1" target="_self" title="WELLORD1:th.1">Th1</a></span>;<br/><a NAME="E15:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A15</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="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E16:64_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="ref" href="wellord1.html#T9" target="_self" title="WELLORD1:th.9">Th13</a></span>;<br/><a NAME="E17:64_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E18:64_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="c3">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</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> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E8:64_1"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="txt" href="#E11:64_1"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th43</a></span>;<br/><span class="kw">hence </span><a NAME="E19:64_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="txt" href="#E15:64_1"><span class="lab"><font color="Green" title="E71">A15</font></span></a>, <a class="txt" href="#E7:64_1"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="wellord1.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:64"/>
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="wellord1.html#T25" target="_self" title="WELLORD1:th.25">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:64"/>
( <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  or  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</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> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Y</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="wellord1.html#T52" target="_self" title="WELLORD1:th.52">Th63</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>
