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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>4</sub></font>;<br/>









<a NAME="E1:36"/><i><font color="Green">E19</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) ) &amp; ( ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) implies <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; ( 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>4</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>1</sub></font> ) ) ) )
 
<b>by </b><i><a class="ref" href="zf_model.html#T20" target="_self">Th20</a></i>;<br/>
<b>thus </b><a NAME="E2:36"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</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>4</sub></font><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E20</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>6</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>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E1:36"><i><font color="Green">E19</font></i></a></i>;<br/>
<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>4</sub></font><b> such that </b><br/><a NAME="E5:36_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>7</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>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:36_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zf_model.html#T20" target="_self">Th20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E6:36_1"/>
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>7</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>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E2:36_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E3:36_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:36_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_model.html#E5:36_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zf_model.html#E2:36_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_model.html#E1:36_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:36_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_model.html#E5:36_1"><i><font color="Green">E21</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>6</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>4</sub></font><b> such that </b><a NAME="E3:36"/><i><font color="Green">E20</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>6</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>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<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>4</sub></font><b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>7</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>5</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" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:36_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="zf_model.html#E5:36"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E3:36_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_model.html#E5:36"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zf_model.html#E1:36_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_model.html#E1:36_2"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:36"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zf_model.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E8:36"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">c<sub>7</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>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_model.html#E5:36"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1" target="_self">|=</a>  <a href="zf_lang.html#K15">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_model.html#E1:36"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_model.html#E7:36"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
