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

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

<b>thus </b><a NAME="E1:8"/>
( ( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies  <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies  <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a> implies  <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2" target="_self">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> ) )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<b>hence </b> <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>
;<br/><br/>


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

<b>assume </b><a NAME="E3:8"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E4:8"/><b>then </b>
 <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8"/>
 <a href="zf_model.html#K2" target="_self">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" target="_self">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>


</div>
