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<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>assume </b><a NAME="E2:66_1_1"/>
not <span class="p1">(<span class="default"><a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:66_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:66_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="closure1.html#E1:66_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="closure1.html#E4:66_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self" title="CLOSURE1:def.13">Def13</a></i>;<br/>
<a NAME="E7:66_1_1"/>
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="closure1.html#E1:66_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></i>;<br/>
<a NAME="E8:66_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="closure1.html#E1:66_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="closure1.html#E6:66_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="closure1.html#E6:66_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="partfun1.html#T54" title="PARTFUN1:th.54">PARTFUN1:54</a></i>;<br/>


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