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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">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="classes2.html#NM1">Universe</a> or  not  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="classes2.html#NM1">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">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="classes2.html#NM1">Universe</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="classes2.html#NM1">Universe</a> ;<br/><b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E3:69_1_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>assume </b><a NAME="E4:69_1_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E5:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#T75">CLASSES1:75</a></i>;<br/><a NAME="E6:69_1_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/><a NAME="E7:69_1_2_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="classes2.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E8:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/><a NAME="E9:69_1_2_1_1"/><b>then </b>
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E7:69_1_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/><b>hence </b><a NAME="E10:69_1_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E6:69_1_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<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">E47</font></i>: 
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is  not   <a href="classes2.html#NM1">Universe</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>2</sub></font> =  <a href="classes1.html#K1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><a NAME="E2:69_1_2_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="classes1.html#T5">CLASSES1:5</a>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/><a NAME="E3:69_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>assume </b><a NAME="E5:69_1_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:69_1_2_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#T75">CLASSES1:75</a></i>;<br/><a NAME="E7:69_1_2_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="classes2.html#T32" target="_self">Th32</a>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/><a NAME="E8:69_1_2_1_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="classes1.html#T43">CLASSES1:43</a>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/><a NAME="E9:69_1_2_1_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E10:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E1:69_1_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="classes2.html#E6:69_1_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="classes2.html#E7:69_1_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E11:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E12:69_1_2_1_2"/><b>then </b>
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E7:69_1_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="classes1.html#T42">CLASSES1:42</a></i>;<br/><a NAME="E13:69_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="classes1.html#R1">is_Tarski-Class_of</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#D3">CLASSES1:def 3</a></i>;<br/><b>hence </b><a NAME="E14:69_1_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#D4">CLASSES1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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