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<b>let </b><font color="Maroon" title="c1">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="c2">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>let </b><font color="Maroon" title="c3">M</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="c4">v</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="c3">M</font>;<br/>







<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> implies  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">v'</font> being   <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="c3">M</font><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green" title="E27">A1</font></i>: 
( ( for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c5">v'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> ) &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">v'</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">v'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
;<br/>

<b>thus </b><a NAME="E4:66_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:66_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="zf_lang1.html#D1" target="_self" title="ZF_LANG1:def.1">Def1</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c5">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font><b> such that </b><a NAME="E3:66"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c6">v'</font> = <font color="Maroon" title="c4">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">m</font>;<br/><b>thus </b><a NAME="E1:66_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c6">v'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#D1" target="_self" title="ZF_LANG1:def.1">Def1</a></i>;<br/><b>thus </b><a NAME="E2:66_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">v'</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:66"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:66"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T20" title="ZF_MODEL:th.20">ZF_MODEL:20</a></i>;<br/>


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