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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> ;<br/>
<a NAME="E1:156"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:156_1"/><i><font color="Green">E62</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T167" target="_self">Th167</a></i>;<br/>
<a NAME="E2:156_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T169" target="_self">Th169</a></i>;<br/>
<b>hence </b><a NAME="E3:156_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:156_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:156"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:156_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> as   <a href="zf_lang.html#NM2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:156_2"/>
 <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T170" target="_self">Th170</a></i>;<br/>
<b>hence </b><a NAME="E4:156_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:156"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E1:156_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> as   <a href="zf_lang.html#NM2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:156_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E175"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:156_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:156"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<b>assume </b><a NAME="E1:156_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> as   <a href="zf_lang.html#NM2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:156_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E5:156_4"/><i><font color="Green">E65</font></i>: 
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>
<a NAME="E6:156_4"/>
 <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E5:156_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_lang1.html#E176"><i><font color="Green">Lemma59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:156_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:156"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" target="_self">ZF_LANG1:sch 1</a>(<a class="txt" href="zf_lang1.html#E1:156"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_lang1.html#E2:156"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_lang1.html#E3:156"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_lang1.html#E4:156"><i><font color="Green">E64</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:156"/>
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 ;<br/>


</div>
