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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<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">b<sub>1</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E3:3_1"/><i><font color="Green">E3</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>6</sub></font>(),<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>())</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V4">negative</a> implies ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>3</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#K20">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V5">conjunctive</a> implies ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</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#K21">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<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">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="zf_lang.html#V6">universal</a> implies ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>3</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#K24">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zf_model.html#E1:3"><i><font color="Green">E1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></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<sub>6</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></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<sub>6</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4">negative</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<a NAME="E2:3_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<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#K20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))
 <b>by </b><i><a class="txt" href="zf_model.html#E1:3"><i><font color="Green">E1</font></i></a>, <a class="txt" href="zf_model.html#E2:3_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5">conjunctive</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_1_2"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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#K21">the_left_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<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="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:3_1_2"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) )
 ;<br/>

<a NAME="E5:3_1_2"/>
( <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:3"><i><font color="Green">E1</font></i></a>, <a class="txt" href="zf_model.html#E2:3_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="zf_model.html#E3:3_1_2"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_2"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>)
 <b>by </b><i><a class="txt" href="zf_model.html#E3:3_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E9:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></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<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_3"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E2:3_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></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<sub>6</sub></font>()</span>)</span>,<font color="Olive">b<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#K24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_3"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></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<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))
 <b>by </b><i><a class="txt" href="zf_model.html#E1:3"><i><font color="Green">E1</font></i></a>, <a class="txt" href="zf_model.html#E2:3_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="zf_model.html#E4:3_1"><i><font color="Green">E4</font></i></a></i>;<br/>


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


</div>
