<?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><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E11</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E12</font></i>: 
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">.</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">.</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E13</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="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">.</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>
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green">E14</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 <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 <b>and </b><br/><a NAME="E5:10"/><i><font color="Green">E15</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund2.html#V1" target="_self">predicatively_closed</a>
 ;<br/>

<a NAME="E6:10"/><i><font color="Green">E16</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a><br/>

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</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>3</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>4</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:10_1"/>
<font color="Maroon">c<sub>5</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_fund2.html#E3:10_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zf_refle.html#D5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</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">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:10_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E4:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E8:10_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</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="E9:10_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E10:10_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E8:10_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/>
<b>hence </b><a NAME="E11:10_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </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>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> , <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_2"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:10_2"/><i><font color="Green">E18</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:10_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="zf_lang.html#K2">x.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zfmodel2.html#T4">ZFMODEL2:4</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span>;<br/><a NAME="E5:10_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 5 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E6:10_2"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:10_2"/><b>then </b><i><font color="Green">E20</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E8:10_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="zf_fund2.html#E2:10_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zf_fund2.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E9:10_2"/><i><font color="Green">E22</font></i>: 
(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="zf_fund2.html#E2:10_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zf_fund2.html#E7:10_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_fund2.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E10:10_2"/>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E11:10_2"/><b>then </b><i><font color="Green">E23</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E8:10_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <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="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/><a NAME="E12:10_2"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</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>1</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="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:10_2_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>10</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>9</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>10</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E3:10_2_1"/>
<font color="Maroon">c<sub>10</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_fund2.html#E2:10_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_refle.html#D5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</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">ZF_REFLE:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E5:10_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="zf_fund2.html#E2:10_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:10_2"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</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>1</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">ZF_REFLE:sch 2</a>(<a class="txt" href="zf_fund2.html#E12:10_2"><i><font color="Green">E24</font></i></a>);<br/></i><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E15:10_2"/>
( <font color="Maroon">c<sub>10</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></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"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a></i>;<br/><a NAME="E16:10_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>10</sub></font> <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;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>10</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>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:10_2"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</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> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>10</sub></font></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="E18:10_2"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <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="10_2_2"><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="E1:10_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 ;<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="E3:10_2_2"/><i><font color="Green">E28</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>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</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#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_fund2.html#E3:10_2_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E5:10_2_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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> &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="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>2</sub></font> 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_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:10_2_2"/>
<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_fund2.html#E3:10_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:10_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><a NAME="E20:10_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E18:10_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:10_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</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_fund2.html#E17:10_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/><a NAME="E22:10_2"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</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="ref" href="zf_refle.html#T28">ZF_REFLE:28</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>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></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>consider </b><font color="Maroon">c<sub>12</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="E25:10_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>12</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>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</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_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E26:10_2"/>
(  <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#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 0 <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="txt" href="zf_fund2.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a>, <a class="ref" href="zf_refle.html#T25">ZF_REFLE:25</a></i>;<br/><a NAME="E27:10_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></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;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E28:10_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></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="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></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>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/><a NAME="E29:10_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</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> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E18:10_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:10_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</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="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/><a NAME="E31:10_2"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</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="ref" href="zf_refle.html#T28">ZF_REFLE:28</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></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/><a NAME="E33:10_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>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="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><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">.</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><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><a NAME="E1:10_2_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>16</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="E2:10_2_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10_2_3"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>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="E5:10_2_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <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">.</a> <font color="Maroon">c<sub>18</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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>18</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_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:10_2_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_fund2.html#E2:10_2_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_fund2.html#E5:10_2_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:10_2_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E8:10_2_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_fund2.html#E1:10_2_3"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:10_2_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E5:10_2_3"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>16</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="E35:10_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E36:10_2"/><i><font color="Green">E31</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>16</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">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="zf_fund2.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_fund2.html#E3:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zf_refle.html#T29">ZF_REFLE:29</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</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="E38:10_2"/><i><font color="Green">E32</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="zf_fund2.html#E35:10_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfrefle1.html#T30">ZFREFLE1:30</a></i>;<br/><a NAME="E39:10_2"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <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/><a NAME="E40:10_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:10_2_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_2_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</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>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="zf_lang.html#NM1">Variable</a> <b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_4"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</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="E6:10_2_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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">.</a> <font color="Maroon">c<sub>21</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>5</sub></font> <a href="funct_2.html#K3">.</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">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>21</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_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:10_2_4"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E14:10_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_fund2.html#E1:10_2_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_fund2.html#E3:10_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund2.html#E6:10_2_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:10_2_4"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <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>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:10_2_4"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E38:10_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E10:10_2_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</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">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E2:10"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:10_2_4"/>
<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">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund2.html#E6:10_2_4"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>5</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">.</a> <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E39:10_2"><i><font color="Green">E33</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="E42:10_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E43:10_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E44:10_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E38:10_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E45:10_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</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">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E2:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_fund2.html#E33:10_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E46:10_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E47:10_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 5 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E48:10_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E49:10_2"/><b>then </b>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/><a NAME="E50:10_2"/><b>then </b><i><font color="Green">E37</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E51:10_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E52:10_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_refle.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E53:10_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E38:10_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E54:10_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E2:10"><i><font color="Green">E12</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>10</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">.</a> <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E25:10_2"><i><font color="Green">E28</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font>;<br/><a NAME="E56:10_2"/><i><font color="Green">E39</font></i>: 
( 10 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 10 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; 10 <a href="xxreal_0.html#R1">&lt;=</a> 10 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> &amp; 10 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E57:10_2"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/><a NAME="E58:10_2"/><i><font color="Green">E41</font></i>: 
(  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T201">ZF_LANG1:201</a></i>;<br/><a NAME="E59:10_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 10 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 5 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 10 )
 ;<br/><a NAME="E60:10_2"/><b>then </b>
( not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E61:10_2"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E62:10_2"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> &amp; (  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> implies  not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/><a NAME="E63:10_2"/><b>then </b>
not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E58:10_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E64:10_2"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E57:10_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E65:10_2"/><b>then </b><i><font color="Green">E42</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E58:10_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E66:10_2"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_5_1_1"/>
(  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">c<sub>4</sub></font> or  not  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_5_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:10_2_5_1_1_1"/>
( not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:10_2_5_1_1_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_fund2.html#E48:10_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zf_fund2.html#E56:10_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/><a NAME="E4:10_2_5_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10_2_5_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zf_fund2.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:10_2_5_1_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:10_2_5_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_2_5_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_5_1_1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E5:10_2_5_1_1_1_1"/>
(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E9:10_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zf_fund2.html#E11:10_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmodel2.html#T11">ZFMODEL2:11</a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E6:10_2_5_1_1_1_1"/><b>then </b>
(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E50:10_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmodel2.html#T14">ZFMODEL2:14</a>, <a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E7:10_2_5_1_1_1_1"/><b>then </b>
(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; not  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/>
<a NAME="E8:10_2_5_1_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="zfmodel2.html#T6">ZFMODEL2:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</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">.</a> <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E46:10_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund2.html#E1:10_2_5_1_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E10:10_2_5_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E4:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E11:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E25:10_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zf_fund2.html#E54:10_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</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">.</a> <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_5_1_1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E13:10_2_5_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E14:10_2_5_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_5_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund2.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/>
<a NAME="E15:10_2_5_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4">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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E16:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E36:10_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_fund2.html#E38:10_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_fund2.html#E8:10_2_5_1_1_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E17:10_2_5_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E65:10_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmodel2.html#T6">ZFMODEL2:6</a>, <a class="ref" href="zfmodel2.html#T9">ZFMODEL2:9</a></i>;<br/>
<a NAME="E18:10_2_5_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E57:10_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmodel2.html#T7">ZFMODEL2:7</a></i>;<br/>
<a NAME="E19:10_2_5_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</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#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</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#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E57:10_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E20:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E3:10_2_5_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E21:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E18:10_2_5_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E22:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T82">ZF_LANG1:82</a></i>;<br/>
<a NAME="E23:10_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E24:10_2_5_1_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2_5_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zf_fund2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:10_2_5_1_1_1"/>
 <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_5_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:10_2_5_1_1_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:10_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2_5_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zf_fund2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</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">.</a> <font color="Maroon">c<sub>17</sub></font><b> such that </b><br/><a NAME="E4:10_2_5_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>25</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">.</a> <font color="Maroon">c<sub>17</sub></font><b> such that </b><br/><a NAME="E6:10_2_5_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2_5_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zf_lang1.html#T82">ZF_LANG1:82</a></i>;<br/>
<a NAME="E7:10_2_5_1_1_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</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#K2">x.</a> 3</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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</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#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E57:10_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_fund2.html#E6:10_2_5_1_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E8:10_2_5_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E9:10_2_5_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<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#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="zfmodel2.html#T9">ZFMODEL2:9</a></i>;<br/>
<a NAME="E10:10_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E57:10_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_fund2.html#E7:10_2_5_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmodel2.html#T7">ZFMODEL2:7</a></i>;<br/>
<a NAME="E11:10_2_5_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E65:10_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmodel2.html#T6">ZFMODEL2:6</a></i>;<br/>
<a NAME="E12:10_2_5_1_1_1_2"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <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">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E14:10_2_5_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E15:10_2_5_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/>
<a NAME="E16:10_2_5_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4">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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span>)</span> &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a>, <a class="ref" href="zf_lang1.html#D2">ZF_LANG1:def 2</a></i>;<br/>
<a NAME="E17:10_2_5_1_1_1_2"/><b>then </b>
(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</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#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E36:10_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zf_fund2.html#E38:10_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_fund2.html#E11:10_2_5_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<a NAME="E18:10_2_5_1_1_1_2"/><b>then </b>
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E50:10_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmodel2.html#T13">ZFMODEL2:13</a></i>;<br/>
<a NAME="E19:10_2_5_1_1_1_2"/><b>then </b>
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>26</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E50:10_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmodel2.html#T6">ZFMODEL2:6</a></i>;<br/>
<a NAME="E20:10_2_5_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>26</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund2.html#E9:10_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zf_fund2.html#E11:10_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="zfmodel2.html#T11">ZFMODEL2:11</a></i>;<br/>
<b>hence </b><a NAME="E21:10_2_5_1_1_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2_5_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="zf_fund2.html#E8:10_2_5_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:10_2_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E5:10_2_5_1_1_1"><i><font color="Green">E45</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="hsNdiv(this)" href="javascript:()" title="10_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_5_1_1_2"/><i><font color="Green">E43</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:10_2_5_1_1_2"/>
( not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:10_2_5_1_1_2"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 <b>by </b><i><a class="txt" href="zf_fund2.html#E4:10_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_fund2.html#E48:10_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zf_fund2.html#E56:10_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/><a NAME="E4:10_2_5_1_1_2"/><b>then </b>
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:10_2_5_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zf_fund2.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E5:10_2_5_1_1_2"/><b>then </b>
 <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zf_fund2.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E6:10_2_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E2:10_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zf_fund2.html#E1:10_2_5_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zf_fund2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_2_5"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 5</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 10</span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E67:10_2"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E5:10"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zf_fund2.html#E42:10_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_fund2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:10"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E6:10"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmodel1.html#T19">ZFMODEL1:19</a></i>;<br/>


</div>
