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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="zf_lang.html#NM2">ZF-formula</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">H</font>,<font color="Olive">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( ( <font color="Olive">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Olive">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Olive">H</font></span>)</span>) ) &amp; ( <font color="Olive">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Olive">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Olive">H</font></span>)</span>) ) &amp; ( <font color="Olive">H</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) ) &amp; ( <font color="Olive">H</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H</font></span>)</span>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) ) &amp; ( <font color="Olive">H</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Olive">H</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> ) ) ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b> ex <font color="Olive">a</font>, <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">a</font>];<br/>
<a NAME="E1:1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V7">atomic</a> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><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="E1:1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  or <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D15">ZF_LANG:def 15</a><br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1"/><i><font color="Green">E32</font></i>: 
( ( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) ) or ( <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) ) )
 ;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">y</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">z</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">t</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">z</font>,<font color="Olive">t</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">z</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">t</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> </span>}</span> ;<br/>

<b>thus </b><a NAME="E4:1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">z</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">t</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">z</font>,<font color="Olive">t</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">z</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">t</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E2:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<a NAME="E3:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">z</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">t</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">z</font>,<font color="Olive">t</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">z</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">t</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_2"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/><a NAME="E2:1_1_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a></i>;<br/><a NAME="E3:1_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a>, <a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a></i>;<br/><a NAME="E4:1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">y</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_3"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/><a NAME="E2:1_1_1_3"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a></i>;<br/><a NAME="E3:1_1_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a>, <a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a></i>;<br/><a NAME="E4:1_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">y</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E8:1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E9:1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">y</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span>  or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">z</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">t</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">z</font>,<font color="Olive">t</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">z</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">t</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>thus </b><a NAME="E10:1_1_1"/>
( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/>

<a NAME="E2:1_1_1_4"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a></i>;<br/>
<a NAME="E3:1_1_1_4"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_1_4_1"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T31">ZF_LANG:31</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1_1_1_4"/><b>then </b>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> or  not <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E6:1_1_1_4"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:1_1_1_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>) )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:1_1_1_4"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a></i>;<br/>
<a NAME="E9:1_1_1_4"/><b>then </b>
(  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T6">ZF_LANG:6</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>)
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


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

<b>thus </b><a NAME="E11:1_1_1"/>
( <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/>

<a NAME="E2:1_1_1_5"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a></i>;<br/>
<a NAME="E3:1_1_1_5"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_1_5_1"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T31">ZF_LANG:31</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1_1_1_5"/><b>then </b>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> or  not <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E6:1_1_1_5"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:1_1_1_5"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>) )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:1_1_1_5"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a></i>;<br/>
<a NAME="E9:1_1_1_5"/><b>then </b>
(  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T7">ZF_LANG:7</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_1_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>)
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


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

<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">y</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E3:1_1_1_6"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E4:1_1_1_6"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="zf_lang.html#T31">ZF_LANG:31</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">z</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">t</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">z</font>,<font color="Olive">t</font>)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">z</font> is   <a href="zf_lang.html#NM1">Variable</a>, <font color="Olive">t</font> is   <a href="zf_lang.html#NM1">Variable</a> : <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">y</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E3:1_1_1_7"/><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E4:1_1_1_7"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_7"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="zf_lang.html#T31">ZF_LANG:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:1_1_1"/>
( ( <font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang.html#T37">ZF_LANG:37</a>, <a class="ref" href="zf_lang.html#T38">ZF_LANG:38</a>, <a class="ref" href="zf_lang.html#T39">ZF_LANG:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V4">negative</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_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="E1:1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>, <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>,<font color="Maroon">a</font>]
 ;<br/>

<b>take </b><font color="Maroon">b</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a</font>);<br/>

<b>take </b><font color="Maroon">X</font> = <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>;<br/>

<b>thus </b><a NAME="E3:1_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<a NAME="E4:1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E6:1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E7:1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:1_1_2"/><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> implies ( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T37">ZF_LANG:37</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_2"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a></i>;<br/>

<b>thus </b><a NAME="E11:1_1_2"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a></i>;<br/>

<b>thus </b><a NAME="E12:1_1_2"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_2_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_2_2_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_2_2_2"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E2:1_1_2_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:1_1_2_2_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_2_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E13:1_1_2"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_2_3"/>
<font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_2_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T38">ZF_LANG:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E4:1_1_2_3"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E14:1_1_2"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_2_4"/>
<font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_2_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T39">ZF_LANG:39</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E4:1_1_2_4"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V5">conjunctive</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><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="E1:1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<b>given </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">A1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A1</font>, <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font>,<font color="Maroon">a1</font>]
 ;<br/>

<b>given </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E3:1_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A2</font>, <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>,<font color="Maroon">a2</font>]
 ;<br/>

<b>take </b><font color="Maroon">a</font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a1</font>,<font color="Maroon">a2</font>);<br/>

<b>take </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="xboole_0.html#K2">\/</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"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>;<br/>

<b>thus </b><a NAME="E4:1_1_3"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:1_1_3_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<a NAME="E5:1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E7:1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E8:1_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:1_1_3"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:1_1_3"/><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> implies ( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T38">ZF_LANG:38</a></i>;<br/>
<a NAME="E12:1_1_3"/><b>then </b><i><font color="Green">E69</font></i>: 
( not <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  &amp; not <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  &amp; not <font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a> &amp; not <font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a>, <a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a>, <a class="ref" href="zf_lang.html#T37">ZF_LANG:37</a>, <a class="ref" href="zf_lang.html#T39">ZF_LANG:39</a></i>;<br/>
<b>thus </b><a NAME="E13:1_1_3"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , , , <a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a></i>;<br/>

<b>thus </b><a NAME="E14:1_1_3"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E15:1_1_3"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_3_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_2_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_2_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_2_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_3_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang.html#T37">ZF_LANG:37</a></i>;<br/>


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

<b>thus </b><a NAME="E16:1_1_3"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_3_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_3_1"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E2:1_1_3_3_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:1_1_3_3_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:1_1_3_3_1"/>
 ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_3_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_3_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_3_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_3_2"/>
 ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_3_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_3_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_3_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_3_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_3_3"/>
 ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_3_3"/>
 ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E17:1_1_3"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_3_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_4_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_4_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_3_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3_4_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_3_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:1_1_3_4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:1_1_3_4_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_3_4"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang.html#T39">ZF_LANG:39</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:1_1"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V6">universal</a> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_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>assume </b><a NAME="E1:1_1_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:1_1_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>, <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>,<font color="Maroon">a</font>]
 ;<br/>

<b>take </b><font color="Maroon">b</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">a</font>);<br/>

<b>take </b><font color="Maroon">X</font> = <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>;<br/>

<b>thus </b><a NAME="E3:1_1_4"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_1"><b>proof </b></a><div class="add">

<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/>



<a NAME="E1:1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<a NAME="E4:1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E6:1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E7:1_1_4"/><b>then </b><i><font color="Green">E87</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:1_1_4"/><i><font color="Green">E88</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> implies ( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:1_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 4
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T39">ZF_LANG:39</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_4"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T35">ZF_LANG:35</a></i>;<br/>

<b>thus </b><a NAME="E11:1_1_4"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F</font></span>)</span>) )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T36">ZF_LANG:36</a></i>;<br/>

<b>thus </b><a NAME="E12:1_1_4"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_4_2"/>
<font color="Maroon">F</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_4_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T37">ZF_LANG:37</a></i>;<br/>
<a NAME="E4:1_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_4_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E13:1_1_4"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_4_3"/>
<font color="Maroon">F</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_4_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang.html#T38">ZF_LANG:38</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E4:1_1_4_3"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">d</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E14:1_1_4"/>
( <font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_4_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">F</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_4_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_4_4_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:1_1_4_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_4_4_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_4_4_2"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E2:1_1_4_4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:1_1_4_4_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_4_4"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E5:1_1"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang.html#S1">ZF_LANG:sch 1</a>(, , , );<br/></i>
<b>hence </b><a NAME="E6:1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Maroon">F<sub>6</sub></font>(),<font color="Olive">a</font>]
 ;<br/>


</div>
