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<b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="zf_lang.html#NM1">Variable</a>;<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">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>











<b>assume </b><a NAME="E1:14"/><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:14_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:14_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:14_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:14"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:14"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:14"/><i><font color="Green">E39</font></i>: 
(  <a href="zf_lang.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_lang.html#K8">All</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> is <a href="zf_lang.html#V6">universal</a> &amp;  <a href="zf_lang.html#K24">the_scope_of</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K23">bound_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a> &amp;  <a href="zf_lang.html#K24">the_scope_of</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K23">bound_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<b>then </b><i><font color="Green">E41</font></i>:  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<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"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>
;<br/>
<a NAME="E7:14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:14"/><b>then </b>
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_lang.html#K16">All</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E12:14"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K16">All</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">H</font>
 ;<br/>


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