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<div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</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>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">E46</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">x2</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">x1</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:156_1"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="zf_lang.html#NM1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y1</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y2</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E2:156_1"/>
 ex <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="zf_lang.html#NM1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">z2</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a></i>;<br/>
<b>hence </b><a NAME="E3:156_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font>] )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T74" target="_self">Th74</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">E51</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">H</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">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

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

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66" target="_self">Th66</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">H</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">E52</font></i>: 
for <font color="Olive">H1</font>, <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">H2</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">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E1:156_3"/>
( <font color="Maroon">H1</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  &amp; <font color="Maroon">H2</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">H1</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>, <font color="Maroon">F2</font> = <font color="Maroon">H2</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">F1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T67" target="_self">Th67</a></i>;<br/>
<b>hence </b><a NAME="E4:156_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</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">E65</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">z</font>,<font color="Olive">H</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">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



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

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E5:156_4"/><i><font color="Green">E66</font></i>: 
( ( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 ;<br/>
<a NAME="E6:156_4"/>
 <a href="zf_lang.html#K8">All</a> <font color="Maroon">s</font>,<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E4:156"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zf_lang1.html#T69" target="_self">Th69</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">z</font>,<font color="Maroon">H</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">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" target="_self">ZF_LANG1:sch 1</a>(<a class="ref" href="zf_lang1.html#T72" target="_self">Th72</a>, <a class="ref" href="zf_lang1.html#T74" target="_self">Th74</a>, , );<br/></i>
<b>hence </b><a NAME="E6:156"/>
<font color="Maroon">H</font> <a href="zf_lang1.html#K5" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 ;<br/>


</div>
