<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <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="Olive" title="b1">M</font><br/>  for <font color="Olive" title="b3">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b4">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="Olive" title="b1">M</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b3">H</font> holds <br/>( <font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b3">H</font> iff <font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b3">H</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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="c2">M</font><br/>  for <font color="Olive" title="b2">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b3">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="c2">M</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b2">H</font> holds <br/>( <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b3">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> iff <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b3">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c2">M</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b1">H</font> holds <br/>( <font color="Maroon" title="c2">M</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">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c2">M</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> holds <br/>( <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/><span class="kw">let </span><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="c2">M</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> implies ( <font color="Maroon" title="c2">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="c4">H</font> iff <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>









<a NAME="E1:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">H</font>) implies <font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">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="c1">x</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/>
<a NAME="E2:6"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">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="c1">x</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">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="c1">x</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" onmouseover="rs('funct_7/T34')" onmouseout="rh()">FUNCT_7:34</a></span>;<br/>
<a NAME="E3:6"/>
( <font color="Maroon" title="c2">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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">H</font>) implies <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:6"/>
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> implies ( <font color="Maroon" title="c2">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="c4">H</font> iff <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="zfmodel2.html#T4" target="_self" onmouseover="rs('zfmodel2/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="funct_7.html#T35" onmouseover="rs('funct_7/T35')" onmouseout="rh()">FUNCT_7:35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
