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<b>let </b><font color="Maroon" title="c1">E</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">f</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">E</font> holds <br/> ( not <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">E</font> iff <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> )</span><br/><b>let </b><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</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">E</font> holds <br/> ( not <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> iff <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</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">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> iff <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> )</span><br/>





<a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E2:16"/><b>then </b>
( <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a>, <a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
 <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K3" title="ZF_MODEL:func.3">VAL</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="zf_model.html#E7"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:16"/>
( not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> implies <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> implies  not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K3" title="ZF_MODEL:func.3">VAL</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#D2" target="_self" title="ZF_MODEL:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1"/>
( not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font> implies <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E5:16"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c1">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font></span><br/>

<b>hence </b><a NAME="E6:16"/>
not <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="zf_model.html#E3:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
