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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</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="E1:46"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a> ;<br/>



<b>assume </b><a NAME="E2:46"/><i><font color="Green">E51</font></i>: 
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b>
 <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:46_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T67">ZF_LANG1:67</a></i>;<br/><a NAME="E4:46_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:46_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then 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>3</sub></font><b> such that </b><br/><a NAME="E3:46_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:46_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:46_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_model.html#T6">ZF_MODEL:6</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:46_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="zf_fund1.html#E3:46_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:46_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then 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>3</sub></font><b> such that </b><br/><a NAME="E4:46_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:46_1_1_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>3</sub></font> st ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</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>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:46_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  st <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> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:46_1_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</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>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E3:46_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:46_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_model.html#D4">ZF_MODEL:def 4</a></i>;<br/>
<a NAME="E4:46_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:46_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="zf_lang1.html#T84">ZF_LANG1:84</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zf_model.html#D4">ZF_MODEL:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:46_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_model.html#T6">ZF_MODEL:6</a></i>;<br/>
<b>hence </b><a NAME="E7:46_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:46_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:46_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:46"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><a NAME="E2:46_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zf_fund1.html#T19" target="_self">Th19</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:46_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E18"><i><font color="Green">Lemma17</font></i></a></i>;<br/><a NAME="E5:46_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:46_1_2"/><b>then </b>
 <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><b>then </b><i><font color="Green">E55</font></i>:  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E22"><i><font color="Green">Lemma21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>
;<br/><i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>
<br/>.= 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E22"><i><font color="Green">Lemma21</font></i></a></i>
;<br/><a NAME="E9:46_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T67">ZF_LANG1:67</a></i>;<br/><a NAME="E10:46_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:46_1_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:46_1_2_3"/><b>then </b><i><font color="Green">E59</font></i>: 
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>3</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:46_1_2_3"/>
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/>
<b>hence </b><a NAME="E4:46_1_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:46_1_2_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46"><i><font color="Green">E50</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> ;<br/><a NAME="E12:46_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_fund1.html#E10:46_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zf_fund1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E13:46_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zf_fund1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E14:46_1_2"/>
<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_2_4"/>
<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_1_2_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:46_1_2_4_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:46_1_2_4_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:46_1_2_4_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:46_1_2_4_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E25"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E7:46_1_2_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_1_2_4_1_1"/><i><font color="Green">E64</font></i>: 
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>3</sub></font> st <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:46_1_2_4_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:46_1_2_4_1_1_1"/>
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:46_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T4">ZF_MODEL:4</a></i>;<br/>
<a NAME="E4:46_1_2_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:46_1_2_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E7:46_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E8:46_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/>
<a NAME="E6:46_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>;<br/>
<a NAME="E7:46_1_2_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E13:46_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:46_1_2_4_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E10:46_1_2_4_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><span class="p0">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:46_1_2_4_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="grfunc_1.html#T18">GRFUNC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E11:46_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2_4_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E1:46_1_2_4_1_1_1"><i><font color="Green">E65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:46_1_2_4_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E6:46_1_2_4_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:46_1_2_4_1_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:46_1_2_4_1_1"/><i><font color="Green">E66</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  st <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:46_1_2_4_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zf_model.html#T6">ZF_MODEL:6</a></i>;<br/>
<a NAME="E6:46_1_2_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:46_1_2_4_1_1_2_1"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:46_1_2_4_1_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:46_1_2_4_1_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="txt" href="zf_fund1.html#E2:46_1_2_4_1_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E5:46_1_2_4_1_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E15"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E6:46_1_2_4_1_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:46_1_2_4_1_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2_4_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2_4_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E5:46_1_2_4_1_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="zf_fund1.html#E17"><i><font color="Green">Lemma16</font></i></a></i>
<br/>.= 
<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> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E5:46_1_2_4_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="zf_fund1.html#E6:46_1_2_4_1_1_2_1"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E5:46_1_2_4_1_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="zf_fund1.html#E17"><i><font color="Green">Lemma16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2_4_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2_4_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1_2_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:46_1_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E6:46_1_2_4_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_fund1.html#E1:46_1_2_4_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E5:46_1_2_4_1_1"><i><font color="Green">E66</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:46_1_2_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E6:46_1_2_4_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:46_1_2_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<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>3</sub></font><b> such that </b><br/><a NAME="E4:46_1_2_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:46_1_2_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E13:46_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:46_1_2_4_2"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<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>3</sub></font><b> such that </b><br/><a NAME="E3:46_1_2_4_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E10:46_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_fund1.html#E1:46_1_2_4_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E25"><i><font color="Green">Lemma24</font></i></a></i>;<br/><a NAME="E4:46_1_2_4_2"/><i><font color="Green">E66</font></i>: 
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>3</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2_4_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:46_1_2_4_2"/>
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/><a NAME="E6:46_1_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:46_1_2_4_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="zf_fund1.html#E24"><i><font color="Green">Lemma23</font></i></a></i>;<br/><a NAME="E7:46_1_2_4_2"/><b>then </b><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zf_model.html#T4">ZF_MODEL:4</a></i>;<br/><a NAME="E8:46_1_2_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_2_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:46_1_2_4_2_1_1"/><i><font color="Green">E68</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:46_1_2_4_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T15">GRFUNC_1:15</a></i>;<br/><a NAME="E4:46_1_2_4_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E5:46_1_2_4_2_1_1"/>
<span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E6:46_1_2_4_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T104">RELAT_1:104</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:46_1_2_4_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2_4_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E3:46_1_2_4_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_fund1.html#E1:46_1_2_4_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="zf_fund1.html#E2:46_1_2_4_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4_2_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="grfunc_1.html#T35">GRFUNC_1:35</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1_2_4_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:46_1_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:46_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E4:46_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E24"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>hence </b><a NAME="E10:46_1_2_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:46_1_2_4_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="zf_model.html#T6">ZF_MODEL:6</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_2_4_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 ;<br/><a NAME="E2:46_1_2_4_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:46_1_2_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:46_1_2_4"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:46_1_2_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p4">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E5:46_1_2_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E15:46_1_2"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:46"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E12:46_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_fund1.html#E13:46_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zf_fund1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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