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<b>let </b><font color="Maroon" title="c1">y</font> be    <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">F</font>, <font color="Olive" title="b2">Ch</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st ( <font color="Olive" title="b1">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) holds <br/> <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">Ch</font>,<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">Ch</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) implies  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">Ch</font>,<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:28"/>
( <font color="Maroon" title="c2">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">Ch</font>,<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>

<a NAME="E2:28"/><b>then </b>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">Ch</font>,<font color="Maroon" title="c1">y</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:28"/>
 <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">Ch</font>,<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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