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<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">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">M</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">v</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/>







<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> holds  <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">x</font>,<font color="Olive">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>;<br/>

<a NAME="E2:65_1"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:65_1"/>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">x</font>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>


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

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> holds  <font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">x</font>,<font color="Olive">m</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v'</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/><b>assume </b><a NAME="E1:65_2"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ;<br/><a NAME="E2:65_2"/><b>then </b>
<font color="Maroon">v'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">v</font> <a href="zf_lang1.html#K1" target="_self">/</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:65_2"/>
<font color="Maroon">M</font>,<font color="Maroon">v'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65"/>
<font color="Maroon">M</font>,<font color="Maroon">v</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/>


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