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<span class="kw">let </span><font color="Maroon" title="c1">x</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">O</font> holds <br/> <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">O</font> implies  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">x</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E1:61"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T32" title="WELLORD2:th.32">WELLORD2:32</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:61"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E3:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T49" target="_self" title="CARD_5:th.49">Th49</a>, <a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a></span>;<br/>
<a NAME="E4:61"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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="#E2:61"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="wellord2.html#T9" title="WELLORD2:th.9">WELLORD2:9</a></span>;<br/>
<a NAME="E5:61"/><span class="kw">then </span>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D2" title="WELLORD2:def.2">WELLORD2:def 2</a></span>;<br/>
<a NAME="E6:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T50" title="WELLORD1:th.50">WELLORD1:50</a></span>;<br/>
<a NAME="E7:61"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T9" title="WELLORD2:th.9">WELLORD2:9</a></span>;<br/>
<a NAME="E8:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="wellord1.html#D9" title="WELLORD1:def.9">WELLORD1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A6</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>
;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="relat_1.html#T208" title="RELAT_1:th.208">RELAT_1:208</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:61"/>
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="funct_4.html#T120" title="FUNCT_4:th.120">FUNCT_4:120</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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