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<b>let </b><font color="Maroon" title="c1">M1</font>, <font color="Maroon" title="c2">M2</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>assume </b><a NAME="E1:13"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c1">M1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M2</font> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">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="c1">M1</font> holds <br/> ( <font color="Maroon" title="c1">M1</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font>,<font color="Maroon" title="c2">M2</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D3" target="_self" title="ZFREFLE1:def.3">ZFREFLE1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D2" target="_self" title="ZFREFLE1:def.2">ZFREFLE1:def 2</a><br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E3:13"/>
( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E13">A3</font></i>: 
for <font color="Olive" title="b1">v1</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="c1">M1</font> holds  <font color="Maroon" title="c1">M1</font>,<font color="Olive" title="b1">v1</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon" title="c4">v2</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="c2">M2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a><br/>

<b>consider </b><font color="Maroon" title="c5">v1</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="c1">M1</font>;<br/>
<a NAME="E3:13_1"/>
<font color="Maroon" title="c1">M1</font>,<font color="Maroon" title="c5">v1</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:13_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E4:13_1"/><b>then </b>
( <font color="Maroon" title="c2">M2</font>,<font color="Maroon" title="c2">M2</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c5">v1</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">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="c3">H</font> holds <br/><font color="Maroon" title="c4">v2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E2:13"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:13_1"/>
<font color="Maroon" title="c2">M2</font>,<font color="Maroon" title="c4">v2</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></i>;<br/>


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

<b>assume </b><a NAME="E4:13"/><i><font color="Green" title="E13">A4</font></i>: 
for <font color="Olive" title="b1">v2</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="c2">M2</font> holds  <font color="Maroon" title="c2">M2</font>,<font color="Olive" title="b1">v2</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon" title="c4">v1</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="c1">M1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a><br/>

<a NAME="E5:13"/>
<font color="Maroon" title="c2">M2</font>,<font color="Maroon" title="c2">M2</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c4">v1</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:13"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon" title="c1">M1</font>,<font color="Maroon" title="c4">v1</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:13"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>


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