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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a><br/>

<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E16</font></i>: 
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_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</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>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:20_1"/><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_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</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>7</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/><a NAME="E4:20_1"/>
 <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="E5:20_1"/><b>then </b><i><font color="Green">E19</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><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="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K2">x.</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K2">x.</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:20_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <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/><b>hence </b><a NAME="E4:20_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E7:20_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><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="tarski.html#K2">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E8:20_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E9:20_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</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#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <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#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p0">(<span class="default"><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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/><a NAME="E10:20_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</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> 1</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> 2</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"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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>6</sub></font><b> such that </b><br/><a NAME="E12:20_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1 holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E13:20_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E14:20_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</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> 2</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</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"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</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>6</sub></font><b> such that </b><br/><a NAME="E16:20_1"/><i><font color="Green">E23</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</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> 4</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> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>;<br/><a NAME="E17:20_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:20_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>14</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" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</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>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:20_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <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> <font color="Maroon">c<sub>15</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E8:20_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:20_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</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> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</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>6</sub></font><b> such that </b><br/><a NAME="E11:20_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <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>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E12:20_1_2"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E13:20_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</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> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</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>6</sub></font><b> such that </b><br/><a NAME="E15:20_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</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>6</sub></font><b> such that </b><br/><a NAME="E17:20_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <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> <font color="Maroon">c<sub>15</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:20_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><b>assume </b><a NAME="E2:20_1_2_1"/>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><a NAME="E3:20_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:20_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> iff <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E20:20_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <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>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E21:20_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E19:20_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</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>6</sub></font><b> such that </b><br/><a NAME="E23:20_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1 holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E24:20_1_2"/>
(  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E25:20_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>21</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> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E21:20_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_1_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E2:20_1_2_2"/><i><font color="Green">E39</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E3:20_1_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1
 <b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E4:20_1_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <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="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:20_1_2_2"/>
( <font color="Maroon">c<sub>21</sub></font> <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> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <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> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E6:20_1_2_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <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="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E6:20_1_2_2"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E15:20_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_1_2_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_2_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1_2_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zfmodel1.html#E6:20_1_2_2"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_1_2_2"><i><font color="Green">E41</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:20_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E25:20_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E28:20_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E29:20_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E30:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E28:20_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/>
<a NAME="E31:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/>
<a NAME="E32:20_1_2"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E33:20_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:20_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:20_1_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 ;<br/>

<a NAME="E2:20_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</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>6</sub></font><b> such that </b><br/><a NAME="E5:20_1_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <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> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E6:20_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E7:20_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</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> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E8:20_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:20_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E10:20_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</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>6</sub></font><b> such that </b><br/><a NAME="E12:20_1_3"/><i><font color="Green">E28</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/>
<a NAME="E13:20_1_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</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> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E14:20_1_3"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E16:20_1_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <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>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</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>6</sub></font><b> such that </b><br/><a NAME="E18:20_1_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <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> <font color="Maroon">c<sub>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:20_1_3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>assume </b><a NAME="E2:20_1_3_1"/>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E3:20_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:20_1_3"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> iff <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E21:20_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</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>6</sub></font><b> such that </b><br/><a NAME="E23:20_1_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1 holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/>
<a NAME="E24:20_1_3"/>
(  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E25:20_1_3"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>19</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> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E21:20_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_1_3_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E2:20_1_3_2"/><i><font color="Green">E38</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E3:20_1_3_2"/><i><font color="Green">E39</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E4:20_1_3_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <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>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E5:20_1_3_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E6:20_1_3_2"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <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> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E7:20_1_3_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green">E37</font></i></a></i>;<br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E23:20_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_3_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_3_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_1_3_2"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_1_3_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_1_3_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_1_3_2"><i><font color="Green">E41</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_1_3_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:20_1_3"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfmodel1.html#E25:20_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E28:20_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E29:20_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E14:20_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfmodel1.html#E27:20_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:20_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E17:20_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E20:20_1"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><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>6</sub></font> 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_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</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="Olive">b<sub>1</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> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>assume </b><a NAME="E2:20_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_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E3:20_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="E4:20_2"/><b>then </b><i><font color="Green">E18</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:20_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><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="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:20_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K2">x.</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K2">x.</a> 2 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:20_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <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/><b>hence </b><a NAME="E4:20_2_1"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E6:20_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><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="tarski.html#K2">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E7:20_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</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>7</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:20_2_2_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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>6</sub></font><b> such that </b><br/><a NAME="E4:20_2_2_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 2 holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <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_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:20_2_2_1_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 ;<br/><a NAME="E2:20_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:20_2_2_1_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>13</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" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <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" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</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>6</sub></font><b> such that </b><br/><a NAME="E7:20_2_2_1_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <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>14</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E8:20_2_2_1_1_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</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> 1</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</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>6</sub></font><b> such that </b><br/><a NAME="E11:20_2_2_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <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>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E13:20_2_2_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <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> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_2_2_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><a NAME="E3:20_2_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E4:20_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:20_2_2_1_1_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> implies <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> ) &amp; ( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> implies <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; <font color="Maroon">c<sub>17</sub></font> <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>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E16:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E6:20_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</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>6</sub></font><b> such that </b><br/><a NAME="E18:20_2_2_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1 holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E19:20_2_2_1_1_1_1"/>
(  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E20:20_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</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> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E16:20_2_2_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_2_2_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E3:20_2_2_1_1_1_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:20_2_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>18</sub></font> <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>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <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>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E5:20_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E6:20_2_2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>15</sub></font> <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> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <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> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E7:20_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_1_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1_2"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_1_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_1_2"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_1_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1_2"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_1_2"><i><font color="Green">E35</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E22:20_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E20:20_2_2_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><a NAME="E23:20_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><b>hence </b><a NAME="E24:20_2_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2_2_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</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>6</sub></font><b> such that </b><br/><a NAME="E3:20_2_2_1_1_1_2"/><i><font color="Green">E25</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/><a NAME="E4:20_2_2_1_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</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> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><a NAME="E5:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E6:20_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</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>6</sub></font><b> such that </b><br/><a NAME="E7:20_2_2_1_1_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <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>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</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>6</sub></font><b> such that </b><br/><a NAME="E9:20_2_2_1_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <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> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</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>6</sub></font><b> such that </b><br/><a NAME="E11:20_2_2_1_1_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 1 holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E12:20_2_2_1_1_1_2"/>
(  <a href="zf_lang.html#K14">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E13:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</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> 2</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_2_2_1_1_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_2_1"/><i><font color="Green">E33</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E3:20_2_2_1_1_1_2_1"/><i><font color="Green">E34</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E4:20_2_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <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>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <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>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E5:20_2_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E6:20_2_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>16</sub></font> <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> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <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> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E7:20_2_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>then </b><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2_1"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E5:20_2_2_1_1_1_2_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:20_2_2_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2_1"><i><font color="Green">E34</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:20_2_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E13:20_2_2_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_2_2_1_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E2:20_2_2_1_1_1_2_2"/><i><font color="Green">E34</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1_2_2"><i><font color="Green">E34</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:20_2_2_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1_2_2"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <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>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E15:20_2_2_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmodel1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E18:20_2_2_1_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel1.html#E3:20_2_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel1.html#E4:20_2_2_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><a NAME="E19:20_2_2_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <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#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:20_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <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#R2">in</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:20_2_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel1.html#E9:20_2_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel1.html#E17:20_2_2_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfmodel1.html#E18:20_2_2_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E21:20_2_2_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:20_2_2_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E22:20_2_2_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfmodel1.html#E20:20_2_2_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:20_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</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> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20_2_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/></div><b>end;</b></div><a NAME="E6:20_2_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</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> 4</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> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="zf_lang.html#K2">x.</a> 2 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:20_2_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E7:20_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</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> 2</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</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"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</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> 1</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> 2</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"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><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> 1</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_2_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
( ( 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_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</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> ) iff for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><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>6</sub></font> 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_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</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="Olive">b<sub>1</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> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E2:20"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
