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<b>let </b><font color="Maroon">E</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">sE</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="abian.html#V2" target="_self">covering</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">E</font>;<br/>





<b>assume </b><a NAME="E1:29"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">X</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">sE</font> holds  <font color="Olive">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">X</font>
 ;<br/>

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:29"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="abian.html#R1" target="_self">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D5" target="_self">ABIAN:def 5</a><br/>

<a NAME="E3:29"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:29"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<a NAME="E5:29"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">sE</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:29"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sE</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:29"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E10:29"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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