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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"> ( <font color="Maroon" title="c1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</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="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="c1">X</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<span class="kw">assume </span><a NAME="E3:35"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</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="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="c1">X</font></span><br/>

<a NAME="E4:35"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</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> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c3">x</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>
 ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c5">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="E5:35_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="classes2.html#T41" target="_self" title="CLASSES2:th.41">Th41</a></span>;<br/>
<a NAME="E6:35_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:35"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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="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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="card_1.html#T28" title="CARD_1:th.28">CARD_1:28</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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