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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">n</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<a NAME="E1:68"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E2:68"/><span class="kw">then </span>
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T34" title="CLASSES1:th.34">CLASSES1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:68"/>
(  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">n</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ; <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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