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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> ;<br/><b>let </b><font color="Maroon" title="c2">H</font>, <font color="Maroon" title="c3">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="c4">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>;<br/>







<a NAME="E1:17"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</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="E2:17"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
 <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">E</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">E</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_model.html#E7"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<a NAME="E3:17"/>
<font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="zf_lang.html#T58" title="ZF_LANG:th.58">ZF_LANG:58</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c3">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">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="c3">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="c3">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#T47" title="ZF_LANG:th.47">ZF_LANG:47</a></i>;<br/>
<b>hence </b><a NAME="E5:17"/>
( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c4">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="c3">H'</font>,<font color="Maroon" title="c1">E</font> implies <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span>,<font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E2:17"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<b>assume </b><a NAME="E6:17"/>
<font color="Maroon" title="c4">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"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">H'</font></span>)</span>,<font color="Maroon" title="c1">E</font>
 ;<br/>

<b>hence </b><a NAME="E7:17"/>
( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c4">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="c3">H'</font>,<font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="txt" href="zf_model.html#E2:17"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="zf_model.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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