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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>  st  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">b</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">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="Olive" title="b2">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">a</font></span>)</span> ) holds <br/> ex <font color="Olive" title="b2">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b3">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">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="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font></span>)</span> ) implies  ex <font color="Olive" title="b1">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
( (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">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="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b2">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">phi</font> &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b3">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> holds <br/> for <font color="Olive" title="b4">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b4">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> ) ) );<br/>
<a NAME="E3:28"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  implies  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Olive" title="b1">u</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Olive" title="b1">u</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">H</font> = <font color="Maroon" title="c4">e</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T4" title="ZF_LANG:th.4">ZF_LANG:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E4:28_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/> for <font color="Olive" title="b2">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="zf_refle.html#T20" title="ZF_REFLE:th.20">ZF_REFLE:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">u</font> = <font color="Maroon" title="c6">phi</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c7">u</font>] )</span><br/>

<a NAME="E6:28_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:28_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c7">u</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">phi</font> &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> holds <br/> for <font color="Olive" title="b3">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b3">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">phi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/> for <font color="Olive" title="b2">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:28_1"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/> for <font color="Olive" title="b2">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">Phi</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">Phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">Phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>) )
 <span class="kw">and </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">Phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S5" title="FUNCT_1:sch.5">FUNCT_1:sch 5</a>(<a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E22">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">Phi</font> = <font color="Maroon" title="c4">Phi</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E8:28"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="classes2.html#T61" title="CLASSES2:th.61">CLASSES2:61</a></span>;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T59" title="CLASSES2:th.59">CLASSES2:59</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a>, <a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a>, <a class="ref" href="zf_lang1.html#T134" title="ZF_LANG1:th.134">ZF_LANG1:134</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E12:28"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">and </span><br/><a NAME="E13:28"/>
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E14:28"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p4"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:28"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#T13" target="_self" title="ZFREFLE1:th.13">Th14</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">phi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E16:28"/>
( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E17:28"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E18:28"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span><br/>

<span class="kw">thus </span><a NAME="E19:28"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T16" title="ZF_REFLE:th.16">ZF_REFLE:16</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfrefle1.html#D3" target="_self" title="ZFREFLE1:def.3">ZFREFLE1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span> holds <br/> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span> holds <br/> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> iff  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">va</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> iff  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> )</span><br/>



<a NAME="E20:28"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T4" title="ZF_LANG:th.4">ZF_LANG:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>, <font color="Maroon" title="c11">xi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E22:28"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font>
 <span class="kw">and </span><br/><a NAME="E23:28"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c5">Phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</font>
 <span class="kw">and </span><br/><a NAME="E24:28"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c11">xi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E25:28"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<font color="Maroon" title="c11">xi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E26:28"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/> for <font color="Olive" title="b2">va</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c10">H1</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c10">H1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c11">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E27:28"/>
<font color="Maroon" title="c7">a</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="c11">xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E28:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E29:28"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c12">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>] ) implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">a</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28_2"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E2:28_2"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c12">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">and </span><br/><a NAME="E3:28_2"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Olive" title="b1">b</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>  holds <br/><font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font>
 <span class="kw">and </span><br/><a NAME="E4:28_2"/>
<font color="Maroon" title="c12">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/>

<a NAME="E5:28_2"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c12">a</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="c11">xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E6:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="ordinal2.html#D13" title="ORDINAL2:def.13">ORDINAL2:def 13</a></span>;<br/>
<a NAME="E7:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A24</font></span>: 
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c13">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> or <font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:28_2"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font></span><br/>

<a NAME="E9:28_2"/>
<font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_2"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E10:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A26</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E11:28_2"/>
<font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> = 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E10:28_2"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="txt" href="#E7:28_2"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></span>
<br/>.= 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>

;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E14:28_2"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:28_2"/><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c13">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="ordinal2.html#T21" title="ORDINAL2:th.21">ORDINAL2:21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:28_2"/><span class="lab"><font color="Green" title="E47">A29</font></span>: 
<font color="Maroon" title="c15">e</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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:28_2"/><span class="lab"><font color="Green" title="E48">A30</font></span>: 
<font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28_2"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">e</font> = <font color="Maroon" title="c15">e</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="#E17:28_2"><span class="lab"><font color="Green" title="E47">A29</font></span></a></span>;<br/>
<a NAME="E20:28_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></span>;<br/>
<a NAME="E21:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28_2"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="txt" href="#E17:28_2"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">e</font> = <font color="Maroon" title="c16">e</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></span>;<br/>
<a NAME="E23:28_2"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E10:28_2"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="txt" href="#E17:28_2"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a></span>;<br/>
<a NAME="E24:28_2"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</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="c11">xi</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E25:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A32</font></span>: 
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E26:28_2"/>
<font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c17">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_2"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="txt" href="#E18:28_2"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E27:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28_2"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="txt" href="#E25:28_2"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E28:28_2"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</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="c6">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E29:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A34</font></span>: 
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28_2"><span class="lab"><font color="Green" title="E49">A31</font></span></a>, <a class="ref" href="funct_1.html#T50" title="FUNCT_1:th.50">FUNCT_1:50</a></span>;<br/>
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> = 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>
<br/>.= 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>

;<br/>
<a NAME="E31:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A35</font></span>: 
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28_2"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E32:28_2"/>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E23:28_2"><span class="lab"><font color="Green" title="E49">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:28_2"/>
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:28_2"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="txt" href="#E27:28_2"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:28"/><span class="lab"><font color="Green" title="E38">A36</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">a</font>] implies <font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font>] )</span><br/>

<a NAME="E1:28_3"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">H</font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:28"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E3:28_3"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</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="c11">xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T34" title="ORDINAL4:th.34">ORDINAL4:34</a></span>;<br/>
<a NAME="E4:28_3"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">H</font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">H</font>,<span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E20:28"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="funct_5.html#T38" title="FUNCT_5:th.38">FUNCT_5:38</a></span>;<br/>
<a NAME="E5:28_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_3"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:28_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p3"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A38</font></span>: 
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p4"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E8:28_3"/>
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>,<span class="p4"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:28_3"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">a</font>] implies <font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_3"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:28"/><span class="lab"><font color="Green" title="E39">A39</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
;<br/>
<a NAME="E32:28"/>
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S4" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E39">A39</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E37">A19</font></span></a>);<br/></span>
<a NAME="E33:28"/><span class="kw">then </span>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E29">A11</font></span></a></span>;<br/>
<a NAME="E34:28"/><span class="kw">then </span>
<font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:28"/>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> iff  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c8">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
