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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<a NAME="E1:53"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> ) ) &amp; ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ) &amp; ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> implies ( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) ) &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T49" target="_self">Th49</a>, <a class="ref" href="classes1.html#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E2:53"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> ) iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 ;<br/>


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