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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">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><span class="kw"> such that </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T26" title="WELLORD2:th.26">WELLORD2:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">Q</font> = <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">take </span><font color="Maroon" title="c4">A</font> =  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</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">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/>

<a NAME="E3:9"/>
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="wellord2.html#T25" title="WELLORD2:th.25">WELLORD2:25</a></span>;<br/>
<a NAME="E4:9"/><span class="kw">then </span>
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">A</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D2" title="WELLORD2:def.2">WELLORD2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E6:9"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">X</font>, <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D8" title="WELLORD1:def.8">WELLORD1:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c5">f</font> <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">proj2</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<a NAME="E7:9"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">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> <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">X</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c4">A</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9"/>
( <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c5">f</font> <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">proj2</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="wellord1.html#D7" title="WELLORD1:def.7">WELLORD1:def 7</a>, <a class="ref" href="wellord2.html#T25" title="WELLORD2:th.25">WELLORD2:25</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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