<?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="E1:6"/><i><font color="Green">E4</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/>

<b>thus </b><a NAME="E2:6"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>7</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>;<br/>
<b>assume </b><a NAME="E2:6_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font> holds <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<a NAME="E3:6_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</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="E5:6_1"/><i><font color="Green">E5</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>(  not <font color="Maroon">c<sub>8</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>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> or  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  <a href="zf_lang.html#K2">x.</a> 1 <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>8</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> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</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> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K6">'not'</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#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</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> 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#K11">=&gt;</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> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_model.html#T23">ZF_MODEL:23</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>;<br/>

<a NAME="E6:6_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E8:6_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<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="E10:6_1"/><i><font color="Green">E7</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>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="E11:6_1"/><i><font color="Green">E8</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>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> 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#E10:6_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E12:6_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> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</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> 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></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p0">(<span class="default"><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> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E13:6_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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#K11">=&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#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E11:6_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E14:6_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> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E10:6_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E15:6_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> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E8:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="zfmodel1.html#E10:6_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E16:6_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#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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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#K11">=&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#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:6_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E18:6_1"/><i><font color="Green">E10</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> 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>12</sub></font> <a href="zf_model.html#R1">|=</a> <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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</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> 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#K11">=&gt;</a> <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> 3</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/>
<b>take </b><font color="Maroon">c<sub>13</sub></font> = <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> 3</span>)</span>;<br/>

<a NAME="E19:6_1"/><i><font color="Green">E11</font></i>: 
( <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> 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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span> &amp; <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#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#K11">=&gt;</a> <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> 3</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E18:6_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<b>thus </b><a NAME="E20:6_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <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="E1:6_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:6_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <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#E1:6"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="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#E1:6_1_1"><i><font color="Green">E12</font></i></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="E5:6_1_1"/><i><font color="Green">E13</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> 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>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>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="E6:6_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>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>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> 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#E5:6_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E7:6_1_1"/><b>then </b><i><font color="Green">E14</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"><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#K11">=&gt;</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> 3</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E19:6_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>
<a NAME="E8:6_1_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> 2</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> 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#E18:6_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="zfmodel1.html#E5:6_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E9:6_1_1"/><b>then </b>
<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> 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#E10:6_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zfmodel1.html#E1:6_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zfmodel1.html#E5:6_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E10:6_1_1"/><b>then </b>
<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> 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> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E7:6_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/>
<a NAME="E11:6_1_1"/><b>then </b>
<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#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> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E12:6_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:6_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


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

<a NAME="E21:6_1"/><i><font color="Green">E12</font></i>: 
( <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> 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> 0</span>)</span> &amp; <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#K6">'not'</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#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E19:6_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E22:6_1"/><b>then </b>
not <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> 3</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E23:6_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> 3</span>)</span> <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> <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#E18:6_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/>
<b>hence </b><a NAME="E24:6_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:6_1"><i><font color="Green">E7</font></i></a></i>;<br/>

<a NAME="E25:6_1"/>
( <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> 0</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> 0</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> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E10:6_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zfmodel1.html#E18:6_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E26:6_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E21:6_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><a NAME="E3:6"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E6:6"/>
<font color="Maroon">c<sub>7</sub></font> <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#E1:6"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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#E3:6"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>10</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/>

<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="E9:6"/><i><font color="Green">E7</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> 0</span>)</span> <a href="hidden.html#R1">=</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="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 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>10</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="E10:6"/><i><font color="Green">E8</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> 0 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E9:6"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</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="E12:6"/><i><font color="Green">E9</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> 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>12</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="E13:6"/><i><font color="Green">E10</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> 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#E12:6"><i><font color="Green">E9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font>;<br/><b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E11</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> 2 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_1"/>
<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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 ;<br/><a NAME="E2:6_2_1"/><b>then </b>
( <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> 2</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> 0</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> 0</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> 0</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> 0</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> 0</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="zfmodel1.html#E1:6_2"><i><font color="Green">E11</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>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E4:6_2_1"/><i><font color="Green">E12</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> 2</span>)</span> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:6"><i><font color="Green">E6</font></i></a>, <a class="txt" href="zfmodel1.html#E9:6"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E5:6_2_1"/><i><font color="Green">E13</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> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> &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> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:6_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</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="E7:6_2_1"/><i><font color="Green">E14</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>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="E8:6_2_1"/><i><font color="Green">E15</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>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>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> 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#E7:6_2_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E9:6_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> 0</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> 0</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> 0</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> 0</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> 0</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> 0</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> 2</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> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="zfmodel1.html#E1:6_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="zfmodel1.html#E7:6_2_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E10:6_2_1"/><b>then </b><i><font color="Green">E16</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> 0</span>)</span> &amp; not <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#K4">'='</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#E9:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zfmodel1.html#E4:6_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zfmodel1.html#E7:6_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><a NAME="E11:6_2_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>  <a href="zf_lang.html#K6">'not'</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#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/><a NAME="E12:6_2_1"/><b>then </b><i><font color="Green">E17</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> 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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:6_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</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="E1:6_2_1_1"/><i><font color="Green">E18</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> 4 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_1_1_1"/>
<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> 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="E2:6_2_1_1_1"/><b>then </b>
( <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#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> 2</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> 2</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> 2</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> 2</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> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:6_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="zfmodel1.html#E1:6_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><a NAME="E3:6_2_1_1_1"/><b>then </b>
( <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; <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#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:6_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfmodel1.html#E1:6_2_1_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E4:6_2_1_1_1"/>
<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> 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> 3</span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E7:6_2_1"><i><font color="Green">E14</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="E3:6_2_1_1"/>
<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> 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#K11">=&gt;</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> 3</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/></div><b>end;</b></div><a NAME="E14:6_2_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>  <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#K11">=&gt;</a> <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> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><a NAME="E15:6_2_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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</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> 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#K11">=&gt;</a> <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> 3</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E12:6_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><b>hence </b><a NAME="E16:6_2_1"/>
<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#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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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#K11">=&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#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E8:6_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:6_2"/>
<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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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#K11">=&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#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T18">ZF_MODEL:18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:6"/><b>then </b><i><font color="Green">E11</font></i>: 
<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#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</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> 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></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p0">(<span class="default"><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> 3</span>)</span></span>)</span></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/>
<a NAME="E16:6"/>
<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> 0</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> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12:6"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E17:6"/><b>then </b>
<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> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E3:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="zfmodel1.html#E9:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zfmodel1.html#E12:6"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/>
<a NAME="E18:6"/><b>then </b>
<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> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</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> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K6">'not'</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#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</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> 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#K11">=&gt;</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> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E15:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E19:6"/><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#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</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> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p0">(<span class="default"><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#K11">=&gt;</a> <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> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E13:6"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/>
<a NAME="E20:6"/><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> 0</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> 1</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> 1</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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> 2</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> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&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"><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> 0</span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <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#K4">'='</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang.html#K7">'&amp;'</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> 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#K11">=&gt;</a> <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> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E10:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="zf_model.html#T20">ZF_MODEL:20</a></i>;<br/>
<b>hence </b><a NAME="E21:6"/>
<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_model.html#K9">the_axiom_of_infinity</a> 
 ;<br/>


</div>
