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<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:60"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; (  for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  holds <br/><font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">A</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c2">X</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="c1">L</font> &amp; <font color="Maroon" title="c3">Y</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="c1">L</font> implies <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:60_1"/>
<font color="Maroon" title="c2">X</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="c1">L</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="c3">Y</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="c1">L</font> implies <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:60_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E5:60_1"/>
<font color="Maroon" title="c3">Y</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="c1">L</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="c2">X</font>,<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:60_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E8:60_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">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="#E3:60_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E7:60_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a></span>;<br/><a NAME="E10:60_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E7:60_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E8:60_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a></span>;<br/><a NAME="E11:60_1"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c7">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 ;<br/><a NAME="E12:60_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:60_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E4:60_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/><a NAME="E13:60_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E11:60_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="classes1.html#T37" title="CLASSES1:th.37">CLASSES1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E14:60_1"/>
<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</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="E4:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">L</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></span>;<br/>
<a NAME="E5:60"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="classes1.html#T38" title="CLASSES1:th.38">CLASSES1:38</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E48">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</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="c1">L</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> )</span><br/><span class="kw">assume </span><a NAME="E1:60_2"/>
<font color="Maroon" title="c2">X</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="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:60_2"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c3">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> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E4:60_2"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</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="#E3:60_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a></span>;<br/><a NAME="E6:60_2"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:60_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E4:60_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:60_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:60_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a>, <a class="ref" href="card_3.html#T42" title="CARD_3:th.42">CARD_3:42</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 href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> = 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="classes2.html#E68"><span class="lab"><font color="Green" title="E43">Lm5</font></span></a>, <a class="ref" href="classes2.html#T24" target="_self" title="CLASSES2:th.24">Th25</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>
;<br/>
<a NAME="E8:60"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E6:60"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="card_3.html#T46" title="CARD_3:th.46">CARD_3:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:60"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
