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<span class="kw">let </span><font color="Maroon" title="c1">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"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> iff  ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b2">B</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> iff  ex <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b2">B</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span>
<a NAME="E4:39"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="classes1.html#E40"><span class="lab"><font color="Green" title="E26">Lm2</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> iff  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:39"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> implies  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">B</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">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">B</font> ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">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">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">B</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c4">Y</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<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="E6:39_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E7:39_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:39_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E35">A8</font></span></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">given </span><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><a NAME="E7:39"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E8:39"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">B</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="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E9:39"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E31">A9</font></span></a></span>;<br/>
<a NAME="E10:39"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">B</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:39"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
