<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><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>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E19">A1</font></i>: 
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">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#M1" title="ORDINAL4:mode.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#M1" title="ORDINAL4:mode.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> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_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="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span> ) )
 ;<br/>

<b>set </b><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/>
<a NAME="E2:33"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#D5" target="_self" title="ZF_REFLE:def.5">Def5</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <b>means </b> ex <font color="Olive" title="b1">phi</font> being   <a href="ordinal4.html#M2" title="ORDINAL4:mode.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#M1" title="ORDINAL4:mode.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">f</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> ,<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> 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#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b3">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 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">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 ) ) );<br/>
<a NAME="E4:33"/><i><font color="Green" title="E22">A4</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a>  or <font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</a><br/>

<a NAME="E2:33_1"/><i><font color="Green" title="E24">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">phi</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">phi</font> = <font color="Maroon" title="c5">phi</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">phi</font> = <font color="Maroon" title="c6">phi</font> as    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">phi</font>
;<br/>

<b>thus </b><a NAME="E6:33_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c7">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font> )
 ;<br/>

<a NAME="E2:33_1_1"/><b>then </b>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:33_1_1"/><b>then </b>
( <font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_1"/>
<font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1_1"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:33_1"/>
<font color="Maroon" title="c7">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon" title="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font> &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> )
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c7">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c8">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> = 
<font color="Maroon" title="c7">phi</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">A</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>
<br/>.= 
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">A</font></span>)</span>
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_1.html#T43" title="FUNCT_1:th.43">FUNCT_1:43</a></i>
<br/>.= 
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E4:33_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<a NAME="E5:33_1_2"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<a NAME="E6:33_1_2"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E6:33_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_2"/>
<font color="Maroon" title="c9">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c7">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E8:33_1"/>
( <font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c8">a</font> )
 ;<br/>

<a NAME="E9:33_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">f</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>;<br/>

<b>thus </b><a NAME="E10:33_1"/>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</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="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_3"/><i><font color="Green" title="E27">A13</font></i>: 
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E2:33_1_3"/><i><font color="Green" title="E28">A14</font></i>: 
<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:33_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/>
<div><i><font color="Green" title="E29">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_3_1"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a> 
 ;<br/><a NAME="E2:33_1_3_1"/><b>then </b><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></i>;<br/><a NAME="E3:33_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1_3"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/><b>hence </b><a NAME="E4:33_1_3_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1_3"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_3_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_3_2"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 ;<br/><a NAME="E2:33_1_3_2"/><b>then </b><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a></i>;<br/><a NAME="E3:33_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1_3"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><b>hence </b><a NAME="E4:33_1_3_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_1_3"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_3_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_1_3"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_1_3"><i><font color="Green" title="E29">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E11:33_1"/><i><font color="Green" title="E27">A18</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E12:33_1"/><i><font color="Green" title="E28">A19</font></i>: 
<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:33_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/>
<div><i><font color="Green" title="E29">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_4"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a> 
 ;<br/><a NAME="E2:33_1_4"/><b>then </b><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></i>;<br/><a NAME="E3:33_1_4"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:33_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/><b>hence </b><a NAME="E4:33_1_4"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:33_1"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_4"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_5"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang.html#NR2" title="ZF_LANG:NR.2">is_membership</a> 
 ;<br/><a NAME="E2:33_1_5"/><b>then </b><i><font color="Green" title="E30">A22</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a></i>;<br/><a NAME="E3:33_1_5"/><b>then </b>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:33_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><b>hence </b><a NAME="E4:33_1_5"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:33_1"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_1_5"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:33_1"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="zf_refle.html#E13:33_1"><i><font color="Green" title="E29">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:33"/><i><font color="Green" title="E23">A23</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:33_2"/>
<font color="Maroon" title="c4">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 ;<br/>

<a NAME="E2:33_2"/><b>then </b><i><font color="Green" title="E24">A24</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></i>;<br/>
<b>given </b><font color="Maroon" title="c5">phi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><a NAME="E4:33_2"/><i><font color="Green" title="E25">A25</font></i>: 
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>and </b><br/><a NAME="E5:33_2"/><i><font color="Green" title="E26">A26</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">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">f</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> ,<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> 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#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">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">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ;<br/>

<b>take </b>
<font color="Maroon" title="c5">phi</font>
;<br/>

<b>thus </b><a NAME="E6:33_2"/>
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_2"><i><font color="Green" title="E25">A25</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E7:33_2"/><i><font color="Green" title="E27">A27</font></i>: 
( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">a</font> )
 ;<br/>

<a NAME="E8:33_2"/><i><font color="Green" title="E28">A28</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">f</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>;<br/>

<b>thus </b><a NAME="E9:33_2"/>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</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="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E2:33_2_1"/><b>then </b>
( not  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="zf_refle.html#E8:33_2"><i><font color="Green" title="E28">A28</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E3:33_2_1"/><b>then </b>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_2"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_2"><i><font color="Green" title="E27">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33_2_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E10:33_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E11:33_2"/><b>then </b>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E12:33_2"/><b>then </b>
not  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_2"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_2"><i><font color="Green" title="E27">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:33_2"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:33"/><i><font color="Green" title="E24">A29</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Olive" title="b1">H</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:33_3"/>
<font color="Maroon" title="c4">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 ;<br/>

<a NAME="E2:33_3"/><b>then </b><i><font color="Green" title="E25">A30</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T58" title="ZF_LANG:th.58">ZF_LANG:58</a></i>;<br/>
<b>given </b><font color="Maroon" title="c5">fi1</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><a NAME="E4:33_3"/><i><font color="Green" title="E26">A31</font></i>: 
( <font color="Maroon" title="c5">fi1</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">fi1</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>and </b><br/><a NAME="E5:33_3"/><i><font color="Green" title="E27">A32</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">fi1</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">f</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> ,<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> 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#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">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">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c6">fi2</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><a NAME="E7:33_3"/><i><font color="Green" title="E28">A33</font></i>: 
( <font color="Maroon" title="c6">fi2</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c6">fi2</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>and </b><br/><a NAME="E8:33_3"/><i><font color="Green" title="E29">A34</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">fi2</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">f</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> ,<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> 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#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">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">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ;<br/>

<b>take </b><font color="Maroon" title="c7">phi</font> = <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c5">fi1</font>;<br/>

<a NAME="E9:33_3"/>
 ex <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <br/>( <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c5">fi1</font> &amp; <font color="Olive" title="b1">fi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:33_3"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_3"><i><font color="Green" title="E28">A33</font></i></a>, <a class="ref" href="ordinal4.html#T13" title="ORDINAL4:th.13">ORDINAL4:13</a></i>;<br/>
<b>hence </b><a NAME="E10:33_3"/>
( <font color="Maroon" title="c7">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c7">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_3"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_3"><i><font color="Green" title="E28">A33</font></i></a>, <a class="ref" href="ordinal4.html#T17" title="ORDINAL4:th.17">ORDINAL4:17</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E11:33_3"/><i><font color="Green" title="E30">A35</font></i>: 
( <font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c8">a</font> )
 ;<br/>

<a NAME="E12:33_3"/><i><font color="Green" title="E31">A36</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">fi1</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">fi2</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E13:33_3"/><b>then </b><i><font color="Green" title="E32">A37</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:33_3"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_3"><i><font color="Green" title="E28">A33</font></i></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/>
<div><i><font color="Green" title="E33">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_3_1"/>
<font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font>
 ;<br/><a NAME="E2:33_3_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E13:33_3"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:33_3_1"/><b>then </b><i><font color="Green" title="E34">A39</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:33_3_1"/>
<font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">fi2</font>
 <b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/><a NAME="E5:33_3_1"/><b>then </b><i><font color="Green" title="E35">A40</font></i>: 
<font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E7:33_3"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_3_1"><i><font color="Green" title="E34">A39</font></i></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></i>;<br/><a NAME="E6:33_3_1"/>
<font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:33_3"><i><font color="Green" title="E31">A36</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E7:33_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:33_3"><i><font color="Green" title="E30">A35</font></i></a>, <a class="txt" href="zf_refle.html#E13:33_3"><i><font color="Green" title="E32">A37</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_3_1"><i><font color="Green" title="E35">A40</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:33_3"/><b>then </b><i><font color="Green" title="E34">A41</font></i>: 
<font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E11:33_3"><i><font color="Green" title="E30">A35</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_3"><i><font color="Green" title="E31">A36</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E16:33_3"/><i><font color="Green" title="E35">A42</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">f</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>;<br/>

<b>thus </b><a NAME="E17:33_3"/>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</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="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_3_2"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E2:33_3_2"/><b>then </b>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> &amp;  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_3"><i><font color="Green" title="E25">A30</font></i></a>, <a class="txt" href="zf_refle.html#E16:33_3"><i><font color="Green" title="E35">A42</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E3:33_3_2"/><b>then </b>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_3"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="zf_refle.html#E8:33_3"><i><font color="Green" title="E29">A34</font></i></a>, <a class="txt" href="zf_refle.html#E11:33_3"><i><font color="Green" title="E30">A35</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_3"><i><font color="Green" title="E33">A38</font></i></a>, <a class="txt" href="zf_refle.html#E15:33_3"><i><font color="Green" title="E34">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33_3_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_3"><i><font color="Green" title="E25">A30</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E18:33_3"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E19:33_3"/><b>then </b>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_3"><i><font color="Green" title="E25">A30</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E20:33_3"/><b>then </b>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> &amp;  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_3"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="zf_refle.html#E8:33_3"><i><font color="Green" title="E29">A34</font></i></a>, <a class="txt" href="zf_refle.html#E11:33_3"><i><font color="Green" title="E30">A35</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_3"><i><font color="Green" title="E33">A38</font></i></a>, <a class="txt" href="zf_refle.html#E15:33_3"><i><font color="Green" title="E34">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:33_3"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_3"><i><font color="Green" title="E25">A30</font></i></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><i><font color="Green" title="E25">A43</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:33_4"/>
<font color="Maroon" title="c4">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 ;<br/>

<a NAME="E2:33_4"/><b>then </b><i><font color="Green" title="E26">A44</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></i>;<br/>
<b>given </b><font color="Maroon" title="c5">phi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><a NAME="E4:33_4"/><i><font color="Green" title="E27">A45</font></i>: 
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>and </b><br/><a NAME="E5:33_4"/><i><font color="Green" title="E28">A46</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">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">f</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> ,<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> 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#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">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">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">x0</font> =  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font>;<br/>
<b>set </b><font color="Maroon" title="c7">H'</font> =  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>;<br/>
<div><i><font color="Green" title="E29">A47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_4_1"/><i><font color="Green" title="E30">A48</font></i>: 
not  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:33_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c5">phi</font>
;<br/>

<b>thus </b><a NAME="E1:33_4_1_1"/>
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4"><i><font color="Green" title="E27">A45</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E2:33_4_1_1"/><i><font color="Green" title="E31">A49</font></i>: 
( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c8">a</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">f</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>;<br/>

<b>thus </b><a NAME="E3:33_4_1_1"/>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</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="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_4_1_1_1"/><i><font color="Green" title="E32">A50</font></i>: 
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E2:33_4_1_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<a NAME="E3:33_4_1_1_1"/><b>then </b>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_1_1_1"><i><font color="Green" title="E32">A50</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E4:33_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_4_1_1"><i><font color="Green" title="E31">A49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:33_4_1_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_1"><i><font color="Green" title="E30">A48</font></i></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:33_4_1_1"/><i><font color="Green" title="E32">A51</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E5:33_4_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<a NAME="E6:33_4_1_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_1_1"><i><font color="Green" title="E32">A51</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E7:33_4_1_1"/><b>then </b>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_4_1_1"><i><font color="Green" title="E31">A49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33_4_1_1"/>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_1"><i><font color="Green" title="E30">A48</font></i></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>defpred </b><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> ] <b>means </b> ex <font color="Olive" title="b1">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( $2 <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> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) );<br/>
<a NAME="E7:33_4"/><i><font color="Green" title="E30">A52</font></i>: 
for <font color="Olive" title="b1">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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>  ex <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">h</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_4_2_1_1"/>
( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> or  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_4_2_1_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_4_2_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">h</font>,<font color="Olive" title="b1">a</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_4_2_1_1_2"/><i><font color="Green" title="E31">A53</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_4_2_1_1_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">h</font>,<font color="Olive" title="b1">a</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E2:33_4_2_1_1_2_1"/><i><font color="Green" title="E32">A54</font></i>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c11">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_2_1_1_2"><i><font color="Green" title="E31">A53</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:33_4_2_1_1_2_1"/><i><font color="Green" title="E33">A55</font></i>: 
( <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> &amp; <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">L</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:33_4_2_1_1_2_1"/><i><font color="Green" title="E34">A56</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4_2_1_1_2_1"><i><font color="Green" title="E33">A55</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_2_1_1_2_1"><i><font color="Green" title="E34">A56</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:33_4_2_1_1_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_2_1_1_2_1"><i><font color="Green" title="E34">A56</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c14">x</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">b</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">f</font>
;<br/>

<b>thus </b><a NAME="E10:33_4_2_1_1_2_1"/>
( <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; (  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c15">b</font> <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> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c10">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4_2_1_1_2_1"><i><font color="Green" title="E32">A54</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_2_1_1_2_1"><i><font color="Green" title="E33">A55</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_2_1_1_2_1"><i><font color="Green" title="E34">A56</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_4_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">h</font>,<font color="Olive" title="b1">a</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">rho</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:33_4"/><i><font color="Green" title="E31">A57</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">rho</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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>
 <b>and </b><br/><a NAME="E10:33_4"/><i><font color="Green" title="E32">A58</font></i>: 
for <font color="Olive" title="b1">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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>  ex <font color="Olive" title="b2">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">a</font>] &amp; ( for <font color="Olive" title="b3">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b3">b</font>] holds <br/><font color="Olive" title="b2">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b3">b</font> ) )
 <b>from </b><i><a class="ref" href="zf_refle.html#S1" target="_self" title="ZF_REFLE:sch.1">ZF_REFLE:sch 1</a>(<a class="txt" href="zf_refle.html#E7:33_4"><i><font color="Green" title="E30">A52</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>] <b>means </b>$2 <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"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> $1</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E11:33_4"/><i><font color="Green" title="E33">A59</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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>,<font color="Olive" title="b2">b1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b2</font>] holds <br/><font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b2</font>
 
;<br/>
<a NAME="E12:33_4"/><i><font color="Green" title="E34">A60</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  ex <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>set </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:33_4_3"/><i><font color="Green" title="E35">A61</font></i>: 
<font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_4_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_4_3_1"/><i><font color="Green" title="E36">A62</font></i>: 
( <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">rho</font> &amp; <font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c9">a</font></span>)</span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:33_4_3_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c9">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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<a NAME="E5:33_4_3_1"/><b>then </b>
 <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c9">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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>
 
<b>by </b><i><a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_3_1"><i><font color="Green" title="E36">A62</font></i></a></i>;<br/>
<a NAME="E7:33_4_3_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font> &amp;  ex <font color="Olive" title="b2">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">a</font> <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> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b3">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp; (  ex <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b3">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b4">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b2">b</font> <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> &amp; ( for <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b3">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b4">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">b</font> ) )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4"><i><font color="Green" title="E32">A58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33_4_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_3_1"><i><font color="Green" title="E36">A62</font></i></a>, <a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33_4_3"/>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</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="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</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="c9">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_refle.html#T25" target="_self" title="ZF_REFLE:th.25">Th25</a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<a NAME="E3:33_4_3"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span> &amp;  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</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="c9">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="classes2.html#T67" title="CLASSES2:th.67">CLASSES2:67</a>, <a class="ref" href="funct_5.html#T68" title="FUNCT_5:th.68">FUNCT_5:68</a></i>;<br/>
<a NAME="E4:33_4_3"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c9">a</font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<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="c9">a</font></span>)</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="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T3" title="CARD_2:th.3">CARD_2:3</a>, <a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></i>;<br/>
<a NAME="E5:33_4_3"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c9">a</font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c1">W</font> <a href="classes1.html#NR1" title="CLASSES1:NR.1">is_Tarski-Class</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E6:33_4_3"/><b>then </b>
<font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c9">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_3"><i><font color="Green" title="E35">A61</font></i></a>, <a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a></i>;<br/>
<a NAME="E7:33_4_3"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c9">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T28" target="_self" title="ZF_REFLE:th.28">Th28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">b</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c9">a</font></span>)</span></span>)</span></span>)</span> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">b</font>
;<br/>

<b>thus </b><a NAME="E9:33_4_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">b</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">si</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E14:33_4"/><i><font color="Green" title="E35">A63</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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>,<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="zf_refle.html#S2" target="_self" title="ZF_REFLE:sch.2">ZF_REFLE:sch 2</a>(<a class="txt" href="zf_refle.html#E11:33_4"><i><font color="Green" title="E33">A59</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_4"><i><font color="Green" title="E34">A60</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>) <b>-&gt; </b>   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> $1</span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> $2</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>) <b>-&gt; </b>   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> =  <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> $2,$1;<br/>
<b>consider </b><font color="Maroon" title="c10">ksi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E16:33_4"/><i><font color="Green" title="E36">A64</font></i>: 
<font color="Maroon" title="c10">ksi</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> <font color="Maroon" title="c9">si</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>
 <b>and </b><br/><a NAME="E17:33_4"/><i><font color="Green" title="E37">A65</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon" title="c10">ksi</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E18:33_4"/><i><font color="Green" title="E38">A66</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c10">ksi</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="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font>)
 <b>from </b><i><a class="ref" href="zf_refle.html#S3" target="_self" title="ZF_REFLE:sch.3">ZF_REFLE:sch 3</a>();<br/></i>
<a NAME="E19:33_4"/><i><font color="Green" title="E39">A67</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E20:33_4"/><i><font color="Green" title="E40">A68</font></i>: 
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></i>;<br/>
<a NAME="E21:33_4"/><i><font color="Green" title="E41">A69</font></i>: 
<font color="Maroon" title="c10">ksi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c12">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:33_4_4"/><i><font color="Green" title="E42">A70</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> &amp; <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> )
 ;<br/>

<a NAME="E2:33_4_4"/><b>then </b>
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c12">B</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4"><i><font color="Green" title="E39">A67</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_4"><i><font color="Green" title="E42">A70</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>] <b>means </b>( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> $1 );<br/>
<a NAME="E4:33_4_4"/><i><font color="Green" title="E43">A71</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></i>;<br/>
<a NAME="E5:33_4_4"/><i><font color="Green" title="E44">A72</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">c</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">c</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_4_4_1"/><i><font color="Green" title="E45">A73</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">c</font> implies <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 <b>and </b><br/><a NAME="E2:33_4_4_1"/><i><font color="Green" title="E46">A74</font></i>: 
<font color="Maroon" title="c13">a</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="c15">c</font>
 ;<br/>

<a NAME="E3:33_4_4_1"/><i><font color="Green" title="E47">A75</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">c</font> &amp; <font color="Maroon" title="c10">ksi</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="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E17:33_4"><i><font color="Green" title="E37">A65</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_4_4_1"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:33_4_4_1"/>
( <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_4_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_4_4_1_1_1"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_4_4_1_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font>
 ;<br/><div class="add"><a NAME="E2:33_4_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c15">c</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_4_1"><i><font color="Green" title="E47">A75</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:33_4_4_1_1_1_1"/>
( <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_4_1"><i><font color="Green" title="E45">A73</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_4_4_1_1_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_4_4_1_1_1_2"/>
( <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_4_4_1"/>
<font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</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="c15">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_4_1"><i><font color="Green" title="E47">A75</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:33_4_4"/><i><font color="Green" title="E45">A76</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</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">b</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">b</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_4_4_2"/><i><font color="Green" title="E46">A77</font></i>: 
( <font color="Maroon" title="c15">b</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="c15">b</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:33_4_4_2"/>
for <font color="Olive" title="b1">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">c</font>]
 <b>and </b><br/><a NAME="E3:33_4_4_2"/><i><font color="Green" title="E47">A78</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font>
 ;<br/>

<i><font color="Green" title="E48">A79</font></i>: <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">b</font> = 
 <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>,<font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="zf_refle.html#E18:33_4"><i><font color="Green" title="E38">A66</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_4_2"><i><font color="Green" title="E46">A77</font></i></a></i>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>
<b>by </b><i><a class="ref" href="zf_refle.html#T17" target="_self" title="ZF_REFLE:th.17">Th17</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E5:33_4_4_2"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_4_2"><i><font color="Green" title="E46">A77</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_4_2"><i><font color="Green" title="E47">A78</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E6:33_4_4_2"/><b>then </b><i><font color="Green" title="E49">A80</font></i>: 
<font color="Maroon" title="c10">ksi</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="c13">a</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="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/>
<a NAME="E7:33_4_4_2"/>
<font color="Maroon" title="c10">ksi</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="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E17:33_4"><i><font color="Green" title="E37">A65</font></i></a></i>;<br/>
<a NAME="E8:33_4_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</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="c13">a</font></span>)</span> &amp; <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:33_4_4_2"/><b>then </b>
<font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</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="c13">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E10:33_4_4_2"/>
<font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4_4_2"><i><font color="Green" title="E48">A79</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_4_2"><i><font color="Green" title="E49">A80</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33_4_4"/>
for <font color="Olive" title="b1">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">c</font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S4" target="_self" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="zf_refle.html#E4:33_4_4"><i><font color="Green" title="E43">A71</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_4_4"><i><font color="Green" title="E44">A72</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_4"><i><font color="Green" title="E45">A76</font></i></a>);<br/></i>
<a NAME="E8:33_4_4"/><b>then </b>
<font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c14">b</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_4"><i><font color="Green" title="E42">A70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:33_4_4"/>
<font color="Maroon" title="c10">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">B</font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c9">si</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="c10">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E22:33_4"/><i><font color="Green" title="E42">A81</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:33_4"><i><font color="Green" title="E36">A64</font></i></a></i>;<br/>
<a NAME="E23:33_4"/><i><font color="Green" title="E43">A82</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>4</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="33_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E1:33_4_5"/>
<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ;<br/>

<a NAME="E2:33_4_5"/>
( <font color="Maroon" title="c10">ksi</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="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">si</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="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E17:33_4"><i><font color="Green" title="E37">A65</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:33_4_5"/><b>then </b>
<font color="Maroon" title="c9">si</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="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</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="c11">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:33_4_5"/>
<font color="Maroon" title="c9">si</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="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</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="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:33_4"/><i><font color="Green" title="E44">A83</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E1:33_4_6"/><i><font color="Green" title="E45">A84</font></i>: 
( <font color="Maroon" title="c11">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="c11">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c12">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a><br/>

<b>assume </b><a NAME="E2:33_4_6"/><i><font color="Green" title="E46">A85</font></i>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ;<br/>

<a NAME="E3:33_4_6"/>
<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <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"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c11">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E14:33_4"><i><font color="Green" title="E35">A63</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:33_4_6"/><i><font color="Green" title="E47">A86</font></i>: 
( <font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c11">a</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">B</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4_6"><i><font color="Green" title="E46">A85</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:33_4_6"/><i><font color="Green" title="E48">A87</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">rho</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c11">a</font></span>)</span> &amp; <font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4_6"><i><font color="Green" title="E47">A86</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> <b>by </b><i><a class="txt" href="zf_refle.html#E9:33_4"><i><font color="Green" title="E31">A57</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_4_6"><i><font color="Green" title="E48">A87</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a1</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E10:33_4_6"/><i><font color="Green" title="E49">A88</font></i>: 
( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">h</font> &amp;  ex <font color="Olive" title="b1">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <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> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <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> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4"><i><font color="Green" title="E32">A58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E12:33_4_6"/><i><font color="Green" title="E50">A89</font></i>: 
( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> &amp; (  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <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> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4_6"><i><font color="Green" title="E49">A88</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>;<br/>
<div><i><font color="Green" title="E51">A90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_4_6_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:33_4_6_1"/><b>then </b><i><font color="Green" title="E52">A91</font></i>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:33_4_6_1"/><i><font color="Green" title="E53">A92</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>( <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>given </b><font color="Maroon" title="c20">a</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><b> such that </b><a NAME="E2:33_4_6_1_1"/><i><font color="Green" title="E54">A93</font></i>: 
( <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> &amp; not <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c21">b</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E3:33_4_6_1_1"/><i><font color="Green" title="E55">A94</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font>
 ;<br/>

<a NAME="E4:33_4_6_1_1"/>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b</font> or <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font> or <font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>
<a NAME="E5:33_4_6_1_1"/><b>then </b>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">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="Maroon" title="c21">b</font> or <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</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="Maroon" title="c20">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:33_4_6_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4_6_1_1"><i><font color="Green" title="E54">A93</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_6_1_1"><i><font color="Green" title="E55">A94</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:33_4_6_1"/><i><font color="Green" title="E54">A95</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font>, <font color="Maroon" title="c20">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:33_4_6_1_2"/><i><font color="Green" title="E55">A96</font></i>: 
( ( for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> ) &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> ) )
 ;<br/>

<b>let </b><font color="Maroon" title="c21">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E2:33_4_6_1_2"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">a</font>
 ;<br/>

<a NAME="E3:33_4_6_1_2"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">a</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_6_1_2"><i><font color="Green" title="E55">A96</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33_4_6_1_2"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">a</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_6_1_2"><i><font color="Green" title="E55">A96</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:33_4_6_1"/><i><font color="Green" title="E55">A97</font></i>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c18">z</font>,<font color="Olive" title="b1">y</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S3" title="CARD_3:sch.3">CARD_3:sch 3</a>(<a class="txt" href="zf_refle.html#E2:33_4_6_1"><i><font color="Green" title="E52">A91</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_6_1"><i><font color="Green" title="E53">A92</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_6_1"><i><font color="Green" title="E54">A95</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c19">z</font> = <font color="Maroon" title="c18">z</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_6_1"><i><font color="Green" title="E55">A97</font></i></a></i>;<br/><a NAME="E8:33_4_6_1"/><i><font color="Green" title="E56">A98</font></i>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">s</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="Maroon" title="c11">a</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E7:33_4_6"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_4_6"><i><font color="Green" title="E50">A89</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:33_4_6_1"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:33_4_6_1"/><b>then </b>
( <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">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="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp;  <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="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="zf_refle.html#E20:33_4"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_6"><i><font color="Green" title="E45">A84</font></i></a>, <a class="txt" href="zf_refle.html#E8:33_4_6_1"><i><font color="Green" title="E56">A98</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:33_4_6_1"/><i><font color="Green" title="E57">A99</font></i>: 
( <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> &amp; <font color="Maroon" title="c20">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:33_4_6_1"/><i><font color="Green" title="E58">A100</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:33_4_6_1"><i><font color="Green" title="E57">A99</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:33_4_6_1"/><i><font color="Green" title="E59">A101</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:33_4_6_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/><a NAME="E16:33_4_6_1"/><b>then </b><i><font color="Green" title="E60">A102</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:33_4_6_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E14:33_4_6_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E18:33_4_6_1"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E16:33_4_6_1"><i><font color="Green" title="E60">A102</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c22">x</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/><b>take </b><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c23">x</font>;<br/><b>thus </b><a NAME="E20:33_4_6_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:33_4_6_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="txt" href="zf_refle.html#E15:33_4_6_1"><i><font color="Green" title="E59">A101</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c25">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E21:33_4_6_1"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E22:33_4_6_1"/>
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_6_1"><i><font color="Green" title="E55">A97</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_4_6_1"><i><font color="Green" title="E57">A99</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_4_6_1"><i><font color="Green" title="E58">A100</font></i></a>, <a class="txt" href="zf_refle.html#E15:33_4_6_1"><i><font color="Green" title="E59">A101</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_4_6_2"/><i><font color="Green" title="E52">A103</font></i>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>take </b><font color="Maroon" title="c18">b</font> =  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>;<br/><b>thus </b><a NAME="E2:33_4_6_2"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E20:33_4"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_6"><i><font color="Green" title="E45">A84</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/><b>thus </b><a NAME="E3:33_4_6_2"/>
for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">b</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_6_2"><i><font color="Green" title="E52">A103</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c18">c</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E16:33_4_6"/><i><font color="Green" title="E52">A104</font></i>: 
( <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; ( for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E13:33_4_6"><i><font color="Green" title="E51">A90</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c19">e</font>;<br/>
<b>consider </b><font color="Maroon" title="c20">v</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E19:33_4_6"/><i><font color="Green" title="E53">A105</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/>( ( <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; (  not <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">x</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i>
<a NAME="E20:33_4_6"/><i><font color="Green" title="E54">A106</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">v</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="Maroon" title="c18">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_4_6_3"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">v</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_4_6_3"/><i><font color="Green" title="E55">A107</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">v</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c22">y</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_6_3"><i><font color="Green" title="E55">A107</font></i></a></i>;<br/>
<a NAME="E5:33_4_6_3"/>
( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E6:33_4_6_3"/><b>then </b>
( ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">y</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> ) or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:33_4_6"><i><font color="Green" title="E52">A104</font></i></a>, <a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_6_3"><i><font color="Green" title="E55">A107</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_4_6_3"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c21">v</font> = <font color="Maroon" title="c20">v</font> as   <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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></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#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E22:33_4_6"/><i><font color="Green" title="E55">A108</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<a NAME="E23:33_4_6"/><b>then </b><i><font color="Green" title="E56">A109</font></i>: 
( <font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c18">c</font></span>)</span> &amp; <font color="Maroon" title="c21">v</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font> &amp;  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c18">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E20:33_4_6"><i><font color="Green" title="E54">A106</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a>, <a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">v'</font> = <font color="Maroon" title="c21">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> ;<br/>
<b>consider </b><font color="Maroon" title="c23">a2</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E26:33_4_6"/><i><font color="Green" title="E57">A110</font></i>: 
( <font color="Maroon" title="c23">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v'</font> &amp;  ex <font color="Olive" title="b1">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c22">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c23">a2</font> <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> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c22">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <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> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c23">a2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4"><i><font color="Green" title="E32">A58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">f'</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E28:33_4_6"/><i><font color="Green" title="E58">A111</font></i>: 
( <font color="Maroon" title="c22">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f'</font> &amp; (  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f'</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c23">a2</font> <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> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f'</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E26:33_4_6"><i><font color="Green" title="E57">A110</font></i></a></i>;<br/>
<div><i><font color="Green" title="E59">A112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_4_6_4"/><i><font color="Green" title="E60">A113</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>;<br/><a NAME="E1:33_4_6_4_1"/><i><font color="Green" title="E61">A114</font></i>: 
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_6_4"><i><font color="Green" title="E60">A113</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:33_4_6_4_1_1"/>
<font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_4_6_4_1_1"/><b>then </b>
( <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font> &amp; ( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:33_4_6_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_4_6_4_1"/>
not  <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"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_6_4_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_4_6_4"/>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:33_4_6"><i><font color="Green" title="E50">A89</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="zf_refle.html#E28:33_4_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_6_4"><i><font color="Green" title="E60">A113</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><a NAME="E2:33_4_6_5"/><i><font color="Green" title="E60">A115</font></i>: 
( <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:33_4_6_5_1"/>
<font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_4_6_5_1"/><b>then </b>
( <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font> &amp; ( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:33_4_6_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font>
 ;<br/></div><b>end;</b></div><a NAME="E4:33_4_6_5"/><b>then </b><i><font color="Green" title="E61">A116</font></i>: 
 <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"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4_6_5"><i><font color="Green" title="E60">A115</font></i></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">m'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E6:33_4_6_5"/><i><font color="Green" title="E62">A117</font></i>: 
( <font color="Maroon" title="c26">m'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f'</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="zf_refle.html#E28:33_4_6"><i><font color="Green" title="E58">A111</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_6_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:33_4_6_5_2"/>
<font color="Maroon" title="c27">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_4_6_5_2"/><b>then </b>
( <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m'</font> &amp; ( <font color="Maroon" title="c27">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_6"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:33_4_6_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f'</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E22:33_4_6"><i><font color="Green" title="E55">A108</font></i></a>, <a class="txt" href="zf_refle.html#E28:33_4_6"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E8:33_4_6_5"/><b>then </b>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m'</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_6_5"><i><font color="Green" title="E62">A117</font></i></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></i>;<br/><a NAME="E9:33_4_6_5"/><b>then </b>
( <font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c23">a2</font> &amp; <font color="Maroon" title="c23">a2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">a1</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4_6"><i><font color="Green" title="E49">A88</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_4_6"><i><font color="Green" title="E50">A89</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="zf_refle.html#E26:33_4_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="txt" href="zf_refle.html#E2:33_4_6_5"><i><font color="Green" title="E60">A115</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_6_5"><i><font color="Green" title="E61">A116</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_6_5"><i><font color="Green" title="E62">A117</font></i></a></i>;<br/><b>hence </b><a NAME="E10:33_4_6_5"/>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:33_4_6"/><b>then </b>
( <font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c18">c</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <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"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c18">c</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">si</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">si</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:33_4"><i><font color="Green" title="E31">A57</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_4"><i><font color="Green" title="E35">A63</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_4_6"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="zf_refle.html#E10:33_4_6"><i><font color="Green" title="E49">A88</font></i></a>, <a class="txt" href="zf_refle.html#E12:33_4_6"><i><font color="Green" title="E50">A89</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4_6"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="zf_refle.html#E26:33_4_6"><i><font color="Green" title="E57">A110</font></i></a>, <a class="txt" href="zf_refle.html#E29:33_4_6"><i><font color="Green" title="E59">A112</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E32:33_4_6"/><b>then </b><i><font color="Green" title="E60">A118</font></i>: 
( <font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:33_4_6"><i><font color="Green" title="E52">A104</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:33_4_6"/><b>then </b>
( <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</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="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E34:33_4_6"/><b>then </b>
<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E35:33_4_6"/><b>then </b>
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E32:33_4_6"><i><font color="Green" title="E60">A118</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E36:33_4_6"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4_6"><i><font color="Green" title="E47">A86</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:33_4"/><i><font color="Green" title="E45">A119</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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>4</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_4_7"/><i><font color="Green" title="E46">A120</font></i>: 
( <font color="Maroon" title="c11">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="c11">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:33_4_7"/><i><font color="Green" title="E47">A121</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.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="c11">a</font> holds <br/><font color="Maroon" title="c9">si</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="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font>
 ;<br/>

<b>thus </b><a NAME="E3:33_4_7"/>
<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a><br/>

<b>assume </b><a NAME="E1:33_4_7_1"/><i><font color="Green" title="E48">A122</font></i>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ;<br/>

<a NAME="E2:33_4_7_1"/>
<font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E24:33_4"><i><font color="Green" title="E44">A83</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_7"><i><font color="Green" title="E46">A120</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:33_4_7_1"/><i><font color="Green" title="E49">A123</font></i>: 
( <font color="Maroon" title="c13">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="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <font color="Maroon" title="c12">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">B</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_7_1"><i><font color="Green" title="E48">A122</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:33_4_7_1"/><i><font color="Green" title="E50">A124</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4_7_1"><i><font color="Green" title="E49">A123</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_7_1"><i><font color="Green" title="E50">A124</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:33_4_7_1"/><i><font color="Green" title="E51">A125</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">a</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E9:33_4_7_1"/><b>then </b><i><font color="Green" title="E52">A126</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_7_1"><i><font color="Green" title="E50">A124</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/>
<a NAME="E10:33_4_7_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/>
<a NAME="E12:33_4_7_1"/>
( <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4_7"><i><font color="Green" title="E47">A121</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_7_1"><i><font color="Green" title="E50">A124</font></i></a>, <a class="txt" href="zf_refle.html#E8:33_4_7_1"><i><font color="Green" title="E51">A125</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E13:33_4_7_1"/><b>then </b>
( <font color="Maroon" title="c12">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E21:33_4"><i><font color="Green" title="E41">A69</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_7_1"><i><font color="Green" title="E49">A123</font></i></a>, <a class="txt" href="zf_refle.html#E9:33_4_7_1"><i><font color="Green" title="E52">A126</font></i></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E14:33_4_7_1"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E26:33_4"/><i><font color="Green" title="E46">A127</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S4" target="_self" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="zf_refle.html#E22:33_4"><i><font color="Green" title="E42">A81</font></i></a>, <a class="txt" href="zf_refle.html#E23:33_4"><i><font color="Green" title="E43">A82</font></i></a>, <a class="txt" href="zf_refle.html#E25:33_4"><i><font color="Green" title="E45">A119</font></i></a>);<br/></i>
<a NAME="E27:33_4"/><i><font color="Green" title="E47">A128</font></i>: 
<font color="Maroon" title="c10">ksi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon" title="c12">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:33_4_8"/><i><font color="Green" title="E48">A129</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">A</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4"><i><font color="Green" title="E39">A67</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self" title="ZF_REFLE:th.8">Th8</a></i>;<br/>
<a NAME="E3:33_4_8"/>
<font color="Maroon" title="c11">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_8"><i><font color="Green" title="E48">A129</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:33_4_8"/><b>then </b><i><font color="Green" title="E49">A130</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E21:33_4"><i><font color="Green" title="E41">A69</font></i></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:33_4_8"/><b>then </b><i><font color="Green" title="E50">A131</font></i>: 
(  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_8"><i><font color="Green" title="E48">A129</font></i></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></i>;<br/>
<i><font color="Green" title="E51">A132</font></i>: <font color="Maroon" title="c12">B</font> = 
 <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span>,<font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="txt" href="zf_refle.html#E18:33_4"><i><font color="Green" title="E38">A66</font></i></a>, <a class="txt" href="zf_refle.html#E20:33_4"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_8"><i><font color="Green" title="E48">A129</font></i></a></i>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="ref" href="zf_refle.html#T17" target="_self" title="ZF_REFLE:th.17">Th17</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E7:33_4_8"/><i><font color="Green" title="E52">A133</font></i>: 
<font color="Maroon" title="c12">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a><br/>

<b>assume </b><a NAME="E1:33_4_8_2"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_4_8_2"/><i><font color="Green" title="E53">A134</font></i>: 
( <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> &amp; <font color="Maroon" title="c15">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_8"><i><font color="Green" title="E51">A132</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:33_4_8_2"/><i><font color="Green" title="E54">A135</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">A1</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">X</font> = <font color="Maroon" title="c15">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_8_2"><i><font color="Green" title="E53">A134</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_4_8_2"><i><font color="Green" title="E54">A135</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:33_4_8_2"/>
<font color="Maroon" title="c17">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_8_2"><i><font color="Green" title="E53">A134</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E8:33_4_8_2"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_8_2"><i><font color="Green" title="E53">A134</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:33_4_8"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a><br/>

<b>assume </b><a NAME="E1:33_4_8_3"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:33_4_8_3"/><i><font color="Green" title="E53">A136</font></i>: 
( <font color="Maroon" title="c15">D</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="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> &amp; <font color="Maroon" title="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">D</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:33_4_8_3"/><i><font color="Green" title="E54">A137</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> &amp; <font color="Maroon" title="c15">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_8_3"><i><font color="Green" title="E53">A136</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4_8_3"><i><font color="Green" title="E54">A137</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:33_4_8_3"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_8"><i><font color="Green" title="E48">A129</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_8"><i><font color="Green" title="E49">A130</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_4_8_3"><i><font color="Green" title="E54">A137</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E8:33_4_8_3"/><b>then </b>
( <font color="Maroon" title="c15">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</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="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4_8"><i><font color="Green" title="E49">A130</font></i></a>, <a class="txt" href="zf_refle.html#E5:33_4_8_3"><i><font color="Green" title="E54">A137</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E9:33_4_8_3"/><b>then </b>
<font color="Maroon" title="c15">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4_8"><i><font color="Green" title="E51">A132</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:33_4_8_3"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_8_3"><i><font color="Green" title="E53">A136</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:33_4_8"/>
<font color="Maroon" title="c12">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4_8"><i><font color="Green" title="E50">A131</font></i></a>, <a class="txt" href="zf_refle.html#E7:33_4_8"><i><font color="Green" title="E52">A133</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_4_9"/>
 <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:33_4_9"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c11">gamma</font> = <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c10">ksi</font>;<br/>

<a NAME="E1:33_4_9_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c10">ksi</font> &amp; <font color="Olive" title="b1">f</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="zf_refle.html#E21:33_4"><i><font color="Green" title="E41">A69</font></i></a>, <a class="ref" href="ordinal4.html#T13" title="ORDINAL4:th.13">ORDINAL4:13</a></i>;<br/>
<b>hence </b><a NAME="E2:33_4_9_1"/>
( <font color="Maroon" title="c11">gamma</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c11">gamma</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="zf_refle.html#E21:33_4"><i><font color="Green" title="E41">A69</font></i></a>, <a class="txt" href="zf_refle.html#E27:33_4"><i><font color="Green" title="E47">A128</font></i></a>, <a class="ref" href="ordinal4.html#T17" title="ORDINAL4:th.17">ORDINAL4:17</a></i>;<br/>

<b>let </b><font color="Maroon" title="c12">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E3:33_4_9_1"/><i><font color="Green" title="E48">A138</font></i>: 
( <font color="Maroon" title="c11">gamma</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> <font color="Maroon" title="c12">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="c12">a</font> )
 ;<br/>

<a NAME="E4:33_4_9_1"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">gamma</font> &amp; <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">phi</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">ksi</font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E5:33_4_9_1"/><b>then </b>
( <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span> &amp; <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</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">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="zf_refle.html#E21:33_4"><i><font color="Green" title="E41">A69</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/>
<a NAME="E6:33_4_9_1"/><b>then </b><i><font color="Green" title="E49">A139</font></i>: 
( <font color="Maroon" title="c10">ksi</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c5">phi</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> <font color="Maroon" title="c12">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:33_4_9_1"><i><font color="Green" title="E48">A138</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:33_4_9_1"/><i><font color="Green" title="E50">A140</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">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>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/>
<b>let </b><font color="Maroon" title="c13">f</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>;<br/>

<b>thus </b><a NAME="E8:33_4_9_1"/>
(  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</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="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_4_9_1_1"/><i><font color="Green" title="E51">A141</font></i>: 
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">g</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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>;<br/><b>assume </b><a NAME="E1:33_4_9_1_1_1"/><i><font color="Green" title="E52">A142</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">k</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><a NAME="E1:33_4_9_1_1_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">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>
 <b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a></i>;<br/><a NAME="E2:33_4_9_1_1_1_1"/><b>then </b>
( <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">f</font> &amp; <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c14">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c14">g</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:33_4_9_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> implies  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:33_4_9_1_1_1"><i><font color="Green" title="E52">A142</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:33_4_9_1_1_1"/><b>then </b>
 <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E1:33_4_9_1_1"><i><font color="Green" title="E51">A141</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E4:33_4_9_1_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_9_1"><i><font color="Green" title="E48">A138</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_9_1"><i><font color="Green" title="E49">A139</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_4_9_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E9:33_4_9_1"/><i><font color="Green" title="E51">A143</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <b>and </b><br/><a NAME="E10:33_4_9_1"/><i><font color="Green" title="E52">A144</font></i>: 
not  <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E12:33_4_9_1"/><i><font color="Green" title="E53">A145</font></i>: 
not  <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"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c14">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E10:33_4_9_1"><i><font color="Green" title="E52">A144</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/>
<a NAME="E13:33_4_9_1"/><i><font color="Green" title="E54">A146</font></i>: 
 <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"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c14">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E12:33_4_9_1"><i><font color="Green" title="E53">A145</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E14:33_4_9_1"/><i><font color="Green" title="E55">A147</font></i>: 
( <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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> &amp; <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <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="c12">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">h</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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<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> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a1</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E17:33_4_9_1"/><i><font color="Green" title="E56">A148</font></i>: 
( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">h</font> &amp;  ex <font color="Olive" title="b1">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <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> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</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> , <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <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> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:33_4"><i><font color="Green" title="E32">A58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E19:33_4_9_1"/><i><font color="Green" title="E57">A149</font></i>: 
( <font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <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"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E13:33_4_9_1"><i><font color="Green" title="E54">A146</font></i></a>, <a class="txt" href="zf_refle.html#E17:33_4_9_1"><i><font color="Green" title="E56">A148</font></i></a></i>;<br/>
<a NAME="E20:33_4_9_1"/><i><font color="Green" title="E58">A150</font></i>: 
<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:33_4_9_1"><i><font color="Green" title="E50">A140</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E21:33_4_9_1"/><b>then </b>
( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<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="c12">a</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">si</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#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<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="c12">a</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:33_4"><i><font color="Green" title="E31">A57</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_4"><i><font color="Green" title="E35">A63</font></i></a>, <a class="txt" href="zf_refle.html#E14:33_4_9_1"><i><font color="Green" title="E55">A147</font></i></a>, <a class="txt" href="zf_refle.html#E17:33_4_9_1"><i><font color="Green" title="E56">A148</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E22:33_4_9_1"/><b>then </b>
( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c9">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E26:33_4"><i><font color="Green" title="E46">A127</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E23:33_4_9_1"/><b>then </b>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</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="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:33"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_9_1"><i><font color="Green" title="E49">A139</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">m'</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_9_1"><i><font color="Green" title="E57">A149</font></i></a></i>;<br/>
<a NAME="E25:33_4_9_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <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="Maroon" title="c12">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_9_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_4_9_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:33_4_9_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:33_4_9_1_2"/><i><font color="Green" title="E59">A151</font></i>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c20">y</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E4:33_4_9_1_2"><i><font color="Green" title="E59">A151</font></i></a></i>;<br/>
<a NAME="E6:33_4_9_1_2"/>
( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> )
 
;<br/>
<a NAME="E7:33_4_9_1_2"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m'</font> or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:33_4_9_1"><i><font color="Green" title="E58">A150</font></i></a>, <a class="txt" href="zf_refle.html#E4:33_4_9_1_2"><i><font color="Green" title="E59">A151</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:33_4_9_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">g</font> = <span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> as   <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> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font> <b>by </b><i><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#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E27:33_4_9_1"/>
( <font color="Maroon" title="c19">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m'</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c19">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> ) )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:33_4_9_1"><i><font color="Green" title="E58">A150</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E28:33_4_9_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m'</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E29:33_4_9_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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#K2" title="ZF_LANG1:func.2">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:33_4_9_1"><i><font color="Green" title="E50">A140</font></i></a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E30:33_4_9_1"/><b>then </b>
not  <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#K2" title="ZF_LANG1:func.2">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m'</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E19:33_4_9_1"><i><font color="Green" title="E57">A149</font></i></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E31:33_4_9_1"/><b>then </b>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m'</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E5:33_4"><i><font color="Green" title="E28">A46</font></i></a>, <a class="txt" href="zf_refle.html#E3:33_4_9_1"><i><font color="Green" title="E48">A138</font></i></a>, <a class="txt" href="zf_refle.html#E6:33_4_9_1"><i><font color="Green" title="E49">A139</font></i></a></i>;<br/>
<b>hence </b><a NAME="E32:33_4_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:33_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="txt" href="zf_refle.html#E9:33_4_9_1"><i><font color="Green" title="E51">A143</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E29:33_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:33_4"><i><font color="Green" title="E29">A47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:33"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 <b>from </b><i><a class="ref" href="zf_lang.html#S1" title="ZF_LANG:sch.1">ZF_LANG:sch 1</a>(<a class="txt" href="zf_refle.html#E4:33"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="zf_refle.html#E5:33"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="zf_refle.html#E6:33"><i><font color="Green" title="E24">A29</font></i></a>, <a class="txt" href="zf_refle.html#E7:33"><i><font color="Green" title="E25">A43</font></i></a>);</i><br/>


</div>
