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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">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/><b>let </b><font color="Maroon" title="c5">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/>









<i><font color="Green" title="E26">A1</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c4">m</font></span>)</span> <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> = 
<font color="Maroon" title="c5">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>
<b>by </b><i><a class="ref" href="zf_lang1.html#T78" target="_self" title="ZF_LANG1:th.78">Th78</a></i>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="zf_lang1.html#T78" target="_self" title="ZF_LANG1:th.78">Th78</a></i>
;<br/>
<a NAME="E2:65"/>
for <font color="Olive" title="b1">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><br/> for <font color="Olive" title="b2">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="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> holds <br/><font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">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="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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>let </b><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/>



<b>assume </b><a NAME="E1:65_2"/><i><font color="Green" title="E27">A2</font></i>: 
<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>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">m'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/><a NAME="E1:65_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">m'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">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="c8">m'</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T78" target="_self" title="ZF_LANG1:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E2:65_2_1"/>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c7">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">m'</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#E1:65_2"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="zf_lang1.html#T80" target="_self" title="ZF_LANG1:th.80">Th80</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:65_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">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="c7">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" target="_self" title="ZF_LANG1:th.80">Th80</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:65"/>
( <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>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">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="c4">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:65"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>


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