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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_refle.html#NM2">DOMAIN-Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E20</font></i>: 
(  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:36"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green">E22</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font> ) ) );<br/>
<a NAME="E4:36"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V7">atomic</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D15">ZF_LANG:def 15</a><br/>

<a NAME="E2:36_1"/><i><font color="Green">E25</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E6:36_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:36_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:36_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="ordinal2.html#V3">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:36_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T94">RELAT_1:94</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T43">FUNCT_1:43</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E4:36_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>:  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>
;<br/>
<a NAME="E6:36_1_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font> is <a href="ordinal2.html#V2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E6:36_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E8:36_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E9:36_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E10:36_1"/>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_3"/><i><font color="Green">E28</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:36_1_3"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:36_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/><a NAME="E2:36_1_3_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a></i>;<br/><a NAME="E3:36_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/><b>hence </b><a NAME="E4:36_1_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_3_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/><a NAME="E2:36_1_3_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a></i>;<br/><a NAME="E3:36_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><b>hence </b><a NAME="E4:36_1_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_3_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_1_3"><i><font color="Green">E30</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E11:36_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E12:36_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:36_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/><a NAME="E2:36_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a></i>;<br/><a NAME="E3:36_1_4"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:36_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/><b>hence </b><a NAME="E4:36_1_4"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:36_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/><a NAME="E2:36_1_5"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a></i>;<br/><a NAME="E3:36_1_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:36_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><b>hence </b><a NAME="E4:36_1_5"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:36_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_1_5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:36_1"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zf_refle.html#E13:36_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:36"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V4">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:36_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<a NAME="E2:36_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30">ZF_LANG:def 30</a></i>;<br/>
<b>given </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:36_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E4:36_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E5:36_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_2"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E6:36_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E7:36_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>thus </b><a NAME="E8:36_2"/>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_2_1"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:36_2_1"/><b>then </b>
( not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_refle.html#E7:36_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E3:36_2_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>


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

<b>assume </b><a NAME="E9:36_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E10:36_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E11:36_2"/><b>then </b>
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:36_2"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V5">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:36_3"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<a NAME="E2:36_3"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T58">ZF_LANG:58</a></i>;<br/>
<b>given </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:36_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E4:36_3"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>6</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E5:36_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E6:36_3"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E7:36_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K5">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ordinal2.html#V2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#T13">ORDINAL4:13</a></i>;<br/>
<b>hence </b><a NAME="E8:36_3"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#T17">ORDINAL4:17</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E9:36_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E10:36_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E11:36_3"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:36_3_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E11:36_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:36_3_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:36_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/><a NAME="E5:36_3_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/><a NAME="E6:36_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:36_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E7:36_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_refle.html#E9:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_refle.html#E11:36_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:36_3"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_refle.html#E10:36_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E14:36_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E15:36_3"/>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_3_2"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:36_3_2"/><b>then </b>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zf_refle.html#E14:36_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E3:36_3_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_refle.html#E9:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_refle.html#E13:36_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>


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

<b>assume </b><a NAME="E16:36_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E17:36_3"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E18:36_3"/><b>then </b>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_refle.html#E9:36_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_refle.html#E13:36_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:36_3"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:36"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V6">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:36_4"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E2:36_4"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T62">ZF_LANG:62</a></i>;<br/>
<b>given </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:36_4"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E4:36_4"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_1"/><i><font color="Green">E31</font></i>: 
not  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:36_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E1:36_4_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4"><i><font color="Green">E28</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:36_4_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E3:36_4_1_1"/>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_4_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:36_4_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E3:36_4_1_1_1"/><b>then </b>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E4:36_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_4_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:36_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmodel1.html#T10">ZFMODEL1:10</a></i>;<br/>


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

<b>assume </b><a NAME="E4:36_4_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:36_4_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E6:36_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E4:36_4_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E7:36_4_1_1"/><b>then </b>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_4_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:36_4_1_1"/>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmodel1.html#T10">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">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) );<br/>
<a NAME="E6:36_4"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_4_2_1_1"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> or ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_4_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_4_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_4_2_1_1_2"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:36_4_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:36_4_2_1_1_2_1"/><i><font color="Green">E33</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_2_1_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_4_2_1_1_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_4_2_1_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4_2_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E6:36_4_2_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:36_4_2_1_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_2_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E10:36_4_2_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4_2_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zf_refle.html#E4:36_4_2_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_2_1_1_2_1"><i><font color="Green">E35</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="E4:36_4_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:36_4"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:36_4"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>from </b><i><a class="ref" href="zf_refle.html#S2" target="_self">ZF_REFLE:sch 2</a>(<a class="txt" href="zf_refle.html#E6:36_4"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E10:36_4"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<a NAME="E11:36_4"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:36_4_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_4_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_3_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:36_4_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E5:36_4_3_1"/><b>then </b>
 <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_3_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E7:36_4_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; ( ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>4</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>4</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E9:36_4"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:36_4_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_4_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <a href="zf_lang.html#K1">VAR</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_refle.html#T25" target="_self">Th25</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E3:36_4_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="ordinal1.html#K5">omega</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> &amp;  <a href="fraenkel.html#K1">Funcs</a> <a href="ordinal1.html#K5">omega</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="classes2.html#T67">CLASSES2:67</a>, <a class="ref" href="funct_5.html#T68">FUNCT_5:68</a></i>;<br/>
<a NAME="E4:36_4_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="ordinal1.html#K5">omega</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="ordinal1.html#K5">omega</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a></i>;<br/>
<a NAME="E5:36_4_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E6:36_4_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/>
<a NAME="E7:36_4_3"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T28" target="_self">Th28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E9:36_4_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:36_4"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="zf_refle.html#S3" target="_self">ZF_REFLE:sch 3</a>(<a class="txt" href="zf_refle.html#E10:36_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_refle.html#E11:36_4"><i><font color="Green">E35</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="ordinal2.html#NM1">Ordinal-Sequence</a>) <b>-&gt; </b>   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="zf_refle.html#K2" target="_self">union</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:36_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E16:36_4"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E17:36_4"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="zf_refle.html#S4" target="_self">ZF_REFLE:sch 4</a>();<br/></i>
<a NAME="E18:36_4"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E19:36_4"/><i><font color="Green">E41</font></i>: 
 <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E20:36_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_4_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E2:36_4_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_refle.html#E18:36_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:36_4_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E5:36_4_4"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_4_4_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> implies <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>and </b><br/><a NAME="E2:36_4_4_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E3:36_4_4_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_4_4_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:36_4_4_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_4_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_4_4_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_4_4_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:36_4_4_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:36_4_4_1_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_4_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_4_4_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_4_4_1_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_4_4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_4_4"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_4_4_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:36_4_4_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>and </b><br/><a NAME="E3:36_4_4_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<i><font color="Green">E49</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> = 
 <a href="zf_refle.html#K2" target="_self">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="zf_refle.html#E17:36_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_4_2"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
 <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="zf_refle.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E5:36_4_4_2"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_4_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_4_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E6:36_4_4_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E7:36_4_4_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E8:36_4_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:36_4_4_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E10:36_4_4_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4_4_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_4_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:36_4_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S5" target="_self">ZF_REFLE:sch 5</a>(<a class="txt" href="zf_refle.html#E4:36_4_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_4"><i><font color="Green">E46</font></i></a>);<br/></i>
<a NAME="E8:36_4_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_4"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_4_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E21:36_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="zf_refle.html#E15:36_4"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E22:36_4"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:36_4_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:36_4_5"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="zf_refle.html#K3" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:36_4_5"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:36_4"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:36_4_6"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1" target="_self">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E2:36_4_6"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:36_4_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E13:36_4"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:36_4_6"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:36_4_6"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="zf_refle.html#E8:36_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_refle.html#E7:36_4_6"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:36_4_6"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E9:36_4"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:36_4_6"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:36_4_6"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_6_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:36_4_6_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:36_4_6_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>given </b><font color="Maroon">c<sub>20</sub></font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:36_4_6_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:36_4_6_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<a NAME="E3:36_4_6_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>
<a NAME="E4:36_4_6_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:36_4_6_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_refle.html#E2:36_4_6_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:36_4_6_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:36_4_6_1_2"/><i><font color="Green">E56</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:36_4_6_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<a NAME="E3:36_4_6_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_4_6_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_4_6_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S3">CARD_3:sch 3</a>(<a class="txt" href="zf_refle.html#E2:36_4_6_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_6_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E4:36_4_6_1"><i><font color="Green">E55</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="zf_lang.html#NM1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E6:36_4_6_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E8:36_4_6_1"/><i><font color="Green">E57</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K1">VAR</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E7:36_4_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_4_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:36_4_6_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:36_4_6_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp;  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_refle.html#E19:36_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="zf_refle.html#E8:36_4_6_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:36_4_6_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:36_4_6_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:36_4_6_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E15:36_4_6_1"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:36_4_6_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/><a NAME="E16:36_4_6_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:36_4_6_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E14:36_4_6_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E18:36_4_6_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4_6_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/><b>take </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font>;<br/><b>thus </b><a NAME="E20:36_4_6_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E14:36_4_6_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_refle.html#E15:36_4_6_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E21:36_4_6_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E22:36_4_6_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:36_4_6_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_4_6_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_refle.html#E14:36_4_6_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_refle.html#E15:36_4_6_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_6_2"/><i><font color="Green">E53</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> =  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E2:36_4_6_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><b>thus </b><a NAME="E3:36_4_6_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_2"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:36_4_6"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E13:36_4_6"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E19:36_4_6"/><i><font color="Green">E54</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K1">VAR</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K1">VAR</a>  holds <br/>( ( <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<a NAME="E20:36_4_6"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_4_6_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_6_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="zf_lang.html#NM1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_6_3"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E5:36_4_6_3"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> or  not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E6:36_4_6_3"/><b>then </b>
( ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4_6"><i><font color="Green">E53</font></i></a>, <a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_6_3"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:36_4_6_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E22:36_4_6"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E23:36_4_6"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E20:36_4_6"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a>, <a class="ref" href="funct_5.html#T63">FUNCT_5:63</a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E26:36_4_6"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E9:36_4"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E28:36_4_6"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E26:36_4_6"><i><font color="Green">E58</font></i></a></i>;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_6_4"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:36_4_6_4_1"/><i><font color="Green">E62</font></i>: 
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_4"><i><font color="Green">E61</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:36_4_6_4_1_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:36_4_6_4_1_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; ( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> implies ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_4_6_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_4_6_4_1"/>
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_4_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zf_lang1.html#T84">ZF_LANG1:84</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_4_6_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E12:36_4_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_refle.html#E28:36_4_6"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_6_4"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E1:36_4_6_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:36_4_6_5_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:36_4_6_5_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; ( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> implies ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_4_6_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E3:36_4_6_5"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_6_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zf_lang1.html#T84">ZF_LANG1:84</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:36_4_6_5"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_refle.html#E28:36_4_6"><i><font color="Green">E59</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_6_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:36_4_6_5_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:36_4_6_5_2"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> implies ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:36_4_6_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E22:36_4_6"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_refle.html#E28:36_4_6"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:36_4_6_5"/><b>then </b>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_6_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zf_lang1.html#T84">ZF_LANG1:84</a></i>;<br/><a NAME="E8:36_4_6_5"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E10:36_4_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_4_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_refle.html#E26:36_4_6"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_6_5"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_6_5"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_6_5"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E9:36_4_6_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:36_4_6"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E8:36_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_refle.html#E13:36_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zf_refle.html#E7:36_4_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E10:36_4_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_refle.html#E12:36_4_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_refle.html#E23:36_4_6"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_refle.html#E26:36_4_6"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_refle.html#E29:36_4_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E32:36_4_6"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E16:36_4_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:36_4_6"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E34:36_4_6"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E35:36_4_6"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E32:36_4_6"><i><font color="Green">E61</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E36:36_4_6"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:36_4"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_4_7"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:36_4_7"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:36_4_7"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1" target="_self">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:36_4_7_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:36_4_7_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E23:36_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_7"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E3:36_4_7_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_7_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:36_4_7_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:36_4_7_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_7_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E7:36_4_7_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E9:36_4_7_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E10:36_4_7_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:36_4_7_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E11:36_4_7_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E13:36_4_7_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4_7"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_refle.html#E7:36_4_7_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_refle.html#E9:36_4_7_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E14:36_4_7_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:36_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_7_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_refle.html#E10:36_4_7_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E15:36_4_7_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E25:36_4"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S5" target="_self">ZF_REFLE:sch 5</a>(<a class="txt" href="zf_refle.html#E21:36_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zf_refle.html#E22:36_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="zf_refle.html#E24:36_4"><i><font color="Green">E46</font></i></a>);<br/></i>
<a NAME="E26:36_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="ordinal2.html#V3">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36_4_8"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_refle.html#E18:36_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zf_refle.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:36_4_8"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:36_4_8"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font> is <a href="ordinal2.html#V2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:36_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:36_4_8"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
 <a href="zf_refle.html#K2" target="_self">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="zf_refle.html#E17:36_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="zf_refle.html#E19:36_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_8"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
 <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="zf_refle.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E7:36_4_8"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1" target="_self">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:36_4_8_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_8_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E6:36_4_8"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:36_4_8_2"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_8_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_8_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:36_4_8_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_8_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E8:36_4_8_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_8_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:36_4_8"/>
 <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1" target="_self">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:36_4_8_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:36_4_8_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_4_8_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_8_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_8_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:36_4_8_3"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_8"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E4:36_4_8"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_8_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E8:36_4_8_3"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4_8"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_refle.html#E5:36_4_8_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E9:36_4_8_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E6:36_4_8"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:36_4_8_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_8_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:36_4_8"/>
<font color="Maroon">c<sub>12</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_refle.html#E7:36_4_8"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_9"/>
 <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:36_4_9"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K5">*</a> <font color="Maroon">c<sub>10</sub></font>;<br/>

<a NAME="E1:36_4_9_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K5">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ordinal2.html#V2">increasing</a> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_refle.html#E20:36_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal4.html#T13">ORDINAL4:13</a></i>;<br/>
<b>hence </b><a NAME="E2:36_4_9_1"/>
( <font color="Maroon">c<sub>11</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_refle.html#E20:36_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_refle.html#E26:36_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ordinal4.html#T17">ORDINAL4:17</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:36_4_9_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<a NAME="E4:36_4_9_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E5:36_4_9_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_refle.html#E20:36_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>
<a NAME="E6:36_4_9_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E3:36_4_9_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:36_4_9_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/>

<b>thus </b><a NAME="E8:36_4_9_1"/>
(  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_4_9_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>assume </b><a NAME="E1:36_4_9_1_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><a NAME="E1:36_4_9_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="zf_refle.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E2:36_4_9_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:36_4_9_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> implies  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E1:36_4_9_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:36_4_9_1_1_1"/><b>then </b>
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>14</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E1:36_4_9_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E4:36_4_9_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_9_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_9_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_4_9_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E9:36_4_9_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E10:36_4_9_1"/><i><font color="Green">E53</font></i>: 
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:36_4_9_1"/><i><font color="Green">E54</font></i>: 
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E10:36_4_9_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zf_lang1.html#T80">ZF_LANG1:80</a></i>;<br/>
<a NAME="E13:36_4_9_1"/><i><font color="Green">E55</font></i>: 
 <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E12:36_4_9_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E14:36_4_9_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:36_4_9_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font> holds not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ) ) ) ) holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zf_refle.html#E9:36_4"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E19:36_4_9_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E13:36_4_9_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_refle.html#E17:36_4_9_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E20:36_4_9_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:36_4_9_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E21:36_4_9_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="zf_lang.html#K1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E8:36_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_refle.html#E13:36_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zf_refle.html#E14:36_4_9_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_refle.html#E17:36_4_9_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E22:36_4_9_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E25:36_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E23:36_4_9_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_9_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_9_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E25:36_4_9_1"/>
(  <a href="relset_1.html#K4">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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K1">VAR</a>  &amp;  <a href="relset_1.html#K5">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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_9_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_4_9_1_2"/>
 <a href="relset_1.html#K4">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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K1">VAR</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:36_4_9_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_4_9_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</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" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="zf_lang.html#NM1">Variable</a> <b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4_9_1_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E6:36_4_9_1_2"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> )
 
;<br/>
<a NAME="E7:36_4_9_1_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:36_4_9_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_refle.html#E4:36_4_9_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:36_4_9_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E27:36_4_9_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zf_refle.html#E20:36_4_9_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E28:36_4_9_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E29:36_4_9_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E7:36_4_9_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E30:36_4_9_1"/><b>then </b>
not  <a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" target="_self">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E19:36_4_9_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E31:36_4_9_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_refle.html#E4:36_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zf_refle.html#E3:36_4_9_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_refle.html#E6:36_4_9_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E32:36_4_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:36_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zf_refle.html#E9:36_4_9_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zf_lang1.html#T80">ZF_LANG1:80</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E28:36_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="zf_refle.html#E5:36_4"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:36"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="zf_lang.html#S1">ZF_LANG:sch 1</a>(<a class="txt" href="zf_refle.html#E4:36"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zf_refle.html#E5:36"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zf_refle.html#E6:36"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_refle.html#E7:36"><i><font color="Green">E26</font></i></a>);</i><br/>


</div>
