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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"> for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font>
 ; <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">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<a NAME="E2:84"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:84"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c2">A</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="classes1.html#T78" target="_self" title="CLASSES1:th.78">Th78</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">Y</font>
; <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="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<a NAME="E6:84"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="classes1.html#T76" target="_self" title="CLASSES1:th.76">Th76</a></span>;<br/>
<a NAME="E7:84"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
 <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> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:84"/>
( <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E7:84"><span class="lab"><font color="Green" title="E62">A6</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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