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<b>thus </b><a NAME="E1:9_1_1"/>
not  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> 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="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> &amp;  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:9_1_1_1"/>
<font color="Maroon" title="c2">M</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1_1_1_1"/>
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a> <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>hence </b><a NAME="E2:9_1_1_1"/>
not  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</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>thus </b><a NAME="E2:9_1_1"/>
 <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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="fraenkel.html#D1" title="FRAENKEL:def.1">FRAENKEL:def 1</a> <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or <font color="Maroon" title="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</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">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<b>hence </b><a NAME="E2:9_1_1_2"/>
<font color="Maroon" title="c3">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</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>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_3.html#D10" title="CARD_3:def.10">CARD_3:def 10</a> <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="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or  not <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<b>assume </b><a NAME="E3:9_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E4:9_1_1"/><b>then </b><i><font color="Green" title="E3">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">g</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></i>;<br/>
<b>hence </b> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="closure2.html#E4:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/> <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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