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<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>let </b><font color="Maroon" title="c9">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c10">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c11">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c10">E</font>;<br/>











<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
( <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font> &amp; <font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c11">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">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" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:14_1"/><b>then </b>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:14_1"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:14_1"/>
not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:14"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c11">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="zfmodel1.html#T11" target="_self" title="ZFMODEL1:th.11">Th11</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green" title="E11">A3</font></i>: 
(  <a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">y</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span> &amp;  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">y</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> &amp;  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font> &amp;  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">y</font>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp;  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> &amp;  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">H</font> &amp;  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E12"><i><font color="Green" title="E7">Lm2</font></i></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<b>then </b><i><font color="Green" title="E12">A4</font></i>:  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14" title="ZF_LANG:func.14">All</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="zfmodel1.html#E5:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></i>
;<br/>
<a NAME="E7:14"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:14"/>
<font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c11">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K16" title="ZF_LANG:func.16">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E6:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="zfmodel1.html#T10" target="_self" title="ZFMODEL1:th.10">Th10</a></i>;<br/>


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