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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2_1"/>
(  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> is   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> or  not  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> is   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1"/>
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> is   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> st <br/>( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive" title="b2">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> ) )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">R</font> =  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> as   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> ;<br/><b>take </b>
<font color="Maroon" title="c2">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font> &amp; ( for <font color="Olive" title="b1">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/><b>thus </b><a NAME="E3:69_1_2_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )</span><br/><b>assume </b><a NAME="E4:69_1_2_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E5:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="classes1.html#T75" title="CLASSES1:th.75">CLASSES1:75</a></i>;<br/><a NAME="E6:69_1_2_1_1"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a></i>;<br/><a NAME="E7:69_1_2_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="classes2.html#T32" target="_self" title="CLASSES2:th.32">Th32</a></i>;<br/><a NAME="E8:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/><a NAME="E9:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="classes2.html#E7:69_1_2_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a></i>;<br/><b>hence </b><a NAME="E10:69_1_2_1_1"/>
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="classes2.html#E6:69_1_2_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_2"/><i><font color="Green" title="E48">A4</font></i>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> is  not   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> st <br/>( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive" title="b2">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> ) )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c2">R</font> =  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font> &amp; ( for <font color="Olive" title="b1">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/><a NAME="E2:69_1_2_1_2"/>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T5" title="CLASSES1:th.5">CLASSES1:5</a>, <a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/><a NAME="E3:69_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )</span><br/><b>assume </b><a NAME="E5:69_1_2_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E6:69_1_2_1_2"/><b>then </b><i><font color="Green" title="E49">A5</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="classes1.html#T75" title="CLASSES1:th.75">CLASSES1:75</a></i>;<br/><a NAME="E7:69_1_2_1_2"/><i><font color="Green" title="E50">A6</font></i>: 
(  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="classes2.html#T32" target="_self" title="CLASSES2:th.32">Th32</a>, <a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/><a NAME="E8:69_1_2_1_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a>, <a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/><a NAME="E9:69_1_2_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E10:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="classes2.html#E1:69_1_2_1_2"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="classes2.html#E6:69_1_2_1_2"><i><font color="Green" title="E49">A5</font></i></a>, <a class="txt" href="classes2.html#E7:69_1_2_1_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E11:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E12:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="classes2.html#E7:69_1_2_1_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="classes1.html#T42" title="CLASSES1:th.42">CLASSES1:42</a></i>;<br/><a NAME="E13:69_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c3">Y</font> <a href="classes1.html#R1" title="CLASSES1:pred.1">is_Tarski-Class_of</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#D3" title="CLASSES1:def.3">CLASSES1:def 3</a></i>;<br/><b>hence </b><a NAME="E14:69_1_2_1_2"/>
<font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="classes1.html#D4" title="CLASSES1:def.4">CLASSES1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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